Advanced Topics in Monetary Economics II1 - Study Center

Advanced Topics in Monetary Economics II1

Carl E. Walsh

UC Santa Cruz

September 3-7, 2012

1 c© Carl E. Walsh, 2012.Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 1 / 319

Course outline

New Keynesian monetary models:I State dependent pricing modelsI Gaps and wedges and optimal policy

Uncertainty and the ZLB

Frictions in labor marketsI Search and matching in the labor market

The open economyI Policy in a currency union

Monetary and fiscal interactionsI Fiscal theories versus monetary theories of the price levelI Optimal monetary and fiscal policy

Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 2 / 319

Equilibrium conditions for basic sticky price NK model

Euler condition: C−σt = βRtEt

(PtPt+1

)C−σt+1

MRS = real wage:χNη

t

C−σt

=Wt

Pt

Marginal cost:Wt

Pt=

ϕtZt

Optimal price setting:(p∗tPt

)=

(θ

θ − 1

)HtFt

Aggregate price index: P1−θt = (1−ω)(p∗t )

1−θ +ωP1−θt−1

where Ht = C 1−σt ϕt +ωβEtHt+1; Ft = C 1−σ

t +ωβEtFt+1

Goods market clearing: ∆−1t Yt = Ct

Price dispersion: ∆t =∫ (pt (i)

Pt

)−θ

di ≥ 1


The role of price dispersion

Output is

Yt =∫cjtdj = Zt

∫Njtdj = ZtNt

But

Yt =∫cjtdj = Ct

∫ (pjtPt

)−θ

dj = Ct∆t ⇒ Ct = ∆−1t Yt .

Since ∆t ≥ 1, price dispersion means more has to be produced toachieve a given level of Ct .

I More work effort required to produce a given Ct .


The basic linearized new Keynesian model

1 An expectational IS curve:

xt = Etxt+1 −(1σ

)(it − Etπt+1 − rflext

)2 An inflation adjustment equation:

πt = βEtπt+1 + κxt + et

3 A specification of policy behavior:

it = rflext + φππt + φxxt ; φπ > 1.


Price adjustment: micro factsBils and Klenow (JPE, 2004)

I median duration between price changes is 4.3 months for items in theU.S. CPI with wide variation in frequency across different categories ofgoods and services.

Klenow and Kryvtsov (QJE 2008)I price changes large on average, but a significant fraction of pricechanges are small.

I variations in the size of price changes, rather than variation in thefraction of prices that change, can account for most of the variance ofaggregate inflation.

Nakamura and Steinsson (QJE 2008) —five facts:I sales have a significant effect on estimates of the median durationbetween price changes (US CPI)

F excluding sales increases median duration between changes from 4.5months to 10 months.

I the frequency of price changes follows a seasonal pattern.I the probability the price of an item changes (the hazard function)declines during the first few months after a change in price.


Price adjustmentAlternative approaches

Time dependent pricing modelsI Calvo —fixed probability of adjusting each period.I Taylor’s model of fixed-length contracts

State dependent price modelsI Dotsey, King, and Wolman (1999, 2006)I Dotsey and King (2005)I Golosov and Lucas (2007)I Gertler and Leahy (2008)I Costain and Nakov (2011)


Price adjustment: the basic TDP Calvo model

Each period, the firms that adjust their price are randomly selected: afraction 1−ω of all firms adjust while the remaining ω fraction donot adjust.

I The parameter ω is a measure of the degree of nominal rigidity; alarger ω implies fewer firms adjust each period and the expected timebetween price changes is longer.

For those firms who do adjust their price at time t, they do so tomaximize the expected discounted value of current and future profits.

I Profits at some future date t + s are affected by the choice of price attime t only if the firm has not received another opportunity to adjustbetween t and t + s. The probability of this is ωs .


Price adjustmentThe firm’s decision problem

First consider the firm’s cost minimization problem, which involvesminimizing WtNjt subject to producing cjt = ZtNjt . This problem canbe written as

minNtWtNt + ϕnt (cjt − ZtNjt ) .

where ϕnt is equal to the firm’s nominal marginal cost. The first ordercondition implies

Wt = ϕnt Zt ,

or ϕnt = Wt/Zt . Dividing by Pt yields real marginal cost asϕt = Wt/ (PtZt ).


Price adjustmentThe firm’s decision problem

The firm’s pricing decision problem then involves picking pjt tomaximize

Et∞

∑i=0

ωi∆i ,t+iΠ(pjtPt+i

, ϕt+i , ct+i

)= Et

∞

∑i=0

ωi∆i ,t+i

[(pjtPt+i

)1−θ

−ϕt+i

(pjtPt+i

)−θ]Ct+i ,

where the discount factor ∆i ,t+i is given by βi (Ct+i/Ct )−σ andprofits are

Π(pjt ) =[(

pjtPt+i

)cjt+i − ϕt+icjt+i

]


Price adjustment

All firms adjusting in period t face the same problem, so all adjustingfirms will set the same price.

Let p∗t be the optimal price chosen by all firms adjusting at time t.The first order condition for the optimal choice of p∗t is

Et∞

∑i=0

ωi∆i ,t+i

[(1− θ)

(1pjt

)(p∗tPt+i

)1−θ

+ θϕt+i

(1p∗t

)(p∗tPt+i

)−θ]Ct+i = 0.

Using the definition of ∆i ,t+i ,

(p∗tPt

)=

(θ

θ − 1

) Et ∑∞i=0 ωi βi (Ct+i/Ct )−σ ϕt+i

(Pt+iPt

)θCt+i

Et ∑∞i=0 ωi βi (Ct+i/Ct )−σ

(Pt+iPt

)θ−1Ct+i

.


Price adjustment

The first order condition can be expressed as(p∗tPt

)=HtFt.

where

Ht =(

θ

θ − 1

)Et

∞

∑i=0

ωi βi (Ct+i/Ct )−σ ϕt+i

(Pt+iPt

)θ

Ct+i =(

θ

θ − 1

)ϕtCt +ωβEtHt+1

and

Ft = Et∞

∑i=0

ωi βi (Ct+i/Ct )−σ

(Pt+iPt

)θ−1Ct+i = Ct +ωβEtFt+1


The case of flexible prices

With flexible prices, all firms adjust each period, so ω = 0.

This implies

Ht =(

θ

θ − 1

)ϕtCt ; Ft = Ct

So (p∗tPt

)=HtFt=

(θ

θ − 1

)ϕt = µϕt .

But with all firms setting the same price, p∗t = Pt and

ϕt =Wt/PtZt

=

(θ − 1

θ

)< 1.


The case of sticky prices

When prices are sticky (ω > 0), the firm must take into accountexpected future marginal cost as well as current marginal cost whensetting p∗t .

The aggregate price index is an average of the price charged by thefraction 1−ω of firms setting their price in period t and the averageof the remaining fraction ω of all firms who set prices in earlierperiods.

Because the adjusting firms were selected randomly from among allfirms, the average price of the non-adjusters is just the average priceof all firms that was prevailing in period t − 1.Thus, the average price in period t satisfies

P1−θt = (1−ω)(p∗t )

1−θ +ωP1−θt−1 . (1)


Alternatives: SDP models

SDP models allow price behavior to be influenced by an intensive andan extensive margin

I after a large shock, those firms that adjust will make, on average,bigger adjustments (this is the intensive margin)

I and more firms will adjust (this is the extensive margin).I Size of price changes among firms adjusting can vary and fraction offirms adjusting can vary.

Basic intuition —highway road repairs.I Caplin and Leahy (1987)

Golosov and Lucas (2007) emphasize that firms most likely to adjustare those furthest from their desired price — this is called the selectioneffect.

I The selection effect acts to make the aggregate price level more flexiblethan might be suggested by simply looking at the fraction of firms thatchange price.


Firm specific shocks

Most models of price adjustment developed for use inmacroeconomics have assumed that firms only face aggregate shocks.

This generally implies that all firms that do adjust their price choosethe same new price as they all face the same (aggregate) shock.

The Dotsey, King, and Wolman model features firm-specific shocks tothe menu cost, but these shocks only influence whether a firmadjusts, not how much it changes prices.

In contrast, Golosov and Lucas (2007) and Gertler and Leahy (2008)have emphasized the role of idiosyncratic shocks in influencing whichfirms adjust prices and in generating a distribution of prices acrossfirms.


Assessing models of nominal price rigidities

Klenow and Kryvtsov (QJE 2008)I No model fits all the micro factsI Calvo does surprisingly well except for the declining hazard rate

F Variations in the size of price changes, rather than variation in thefraction of prices that change, can account for most of the variance ofaggregate inflation.

Carlsson and Nordström Skans (AEJ Macro 2012)I Uses Swedish firm-level data to compare models of staggered priceadjustment with models of flexible prices but imperfect information(sticky information, rational inattention to macro factors)

I Concludes Calvo explains data best.


Trend inflationStandard NKPCs in DSGE models add ad hoc assumptions aboutindexation.Example: Justiniano, Primiceri, and Tambalotti (2011) assumenon-optimizing firms and households set

Pt (j) = Pt−1(j)πιpt−1π

1−ιp ; Wt (j) = Wt−1(j) (πt−1ezt )ιw π1−ιw

where π is steady-state inflation and zt is the productivity shock.Estimated values of ιp , ιw ≈ 0 but this still imposes indexation tosteady-state inflation.Costly relative price and wage dispersion depends on

[πt − ιpπt−1 − (1− ιp)π]2 ≈ (πt − π)2

and

[πw ,t − ιw (πt−1 + zt )− (1− ιw )π]2 ≈ (πw ,t − π)2 .

Trend inflation doesn’t matter.Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 18 / 319

Trend and cyclical inflation

1965 1970 1975 1980 1985 1990 1995 2000 2005 20102.5

0.0

2.5

5.0

7.5

10.0

12.5INFLA TION_P CEINFLA TION_TRE NDDE TRE NDE D_INFL

Figure: U.S. inflation (pce), trend inflation (hp), and cyclical inflation


Calvo with trend inflation

Coibion and Gorodnichenko (AER 2011)

πt =

(1−ωΠθ−1

ωΠθ−1

)p∗t

where p∗t is the relative price set by adjusting firms, Π is thesteady-state inflation rate, ω is the Calvo paramter (fraction of firmsnot resetting prices), and θ is the elasticity of substitution acrossgoods.


Calvo with trend inflation

The optimal reset price is

(1+ θη−1

)p∗t =

(1+ η−1

)(1− γ2)

∞

∑j=0

γj2Etxt+j

+Et∞

∑j=0

(γj2 − γj1

)(gt+j − it+j−1)

+Et∞

∑j=0

γj2[1+ θ

(1+ η−1

)]− γj1θ

πt+j

where γ2 = ωR−1gΠθ, γ2 = γ1Π1+θ

η , gt is the growth rate ofoutput, x is the output gap, and i is the nominal interest rate.


Trend inflation

If Calvo model (and micro evidence) taken seriously, one can’t rely onindexation.

Ascari and Ropele (2007), Cogley and Sbordone (2008), Coibion,Gorodnichenko, and Wieland (2011), Alves (2010).

Indexation to trend inflation:

P1−θt = (1−ω) (P∗t )

1−θ +ωP1−θt−1 Πδ(1−θ)

where δ is the degree of indexation.

Steady-state output gap is not equal to zero if δ < 1. Coibion,Gorodnchenko and Wieland show that(

YY f

)1+η

=1−ωβ−1Π(1−δ)θ(1+η)

1−ωβ−1Π(1−δ)(θ−1)

(1−ω

1−ωΠ(1−δ)(θ−1)

) 1+ηθθ−1.


Trend inflation

Steady-state output gap is not equal to zero. Coibion, Gorodnchenkoand Wieland show that

X 1+ηt =

(YY f

)1+η

=1−ωβ−1Π(1−δ)θ(1+η)

1−ωβ−1Π(1−δ)(θ−1)

(1−ω

1−ωΠ(1−δ)(θ−1)

) 1+ηθθ−1.

X is increasing for very low but positive levels of trend inflation butthen decreases.

Optimal rate of inflation is positive but small.

Interestingly, downward wage rigidity lowers the optimal inflation rateand makes ZLB less likely (will return to this point).


Optimal policy: Topics to cover

1 Policy objectives2 Optimal policy under discretion and commitment3 The zero lower bound4 Uncertainty


Policy in forward-looking models: welfareQuadratic approximation

Woodford demonstrates that deviations of the expected discountedutility of the representative agent around the level of steady-stateutility can be approximated by

Et∞

∑i=0

βiVt+i ≈ −12

ΩEt∞

∑i=0

βi[π2t+i + λ (xt+i − x∗)2

]. (2)

xt is the gap between output and the output level that would ariseunder flexible prices, and x∗ is the gap between the steady-stateeffi cient level of output (in the absence of the monopolisticdistortions) and the steady-state level of output.


Policy weights

Theory says something about the weights in the loss function:

Et∞

∑i=0

βiVt+i ≈ −12

ΩEt∞

∑i=0

βi[π2t+i + λ (xt+i − x∗)2

],

where

Ω = Y Uc

[ω

(1−ω)(1−ωβ)

] (θ−1 + η

)θ2

and

λ =

[(1−ω)(1−ωβ)

ω

](σ+ η)

(1+ ηθ) θ.

Greater nominal rigidity (larger ω) reduces λ.

Loss function endogenous.

Calvo specification implies λ is small —Taylor specification leads tolarger weight on output gap.


Policy implications of price stickiness

When the price level fluctuates, and not all firms are able to adjust,price dispersion results. This causes the relative prices of the differentgoods to vary. If the price level rises, for example, two things happen.

1 The relative price of firms who have not set their prices for a whilefalls. They experience in increase in demand and raise output, whilefirms who have just reset their prices reduce output. This productiondispersion is ineffi cient.

2 Consumers increase their consumption of the goods whose relativeprice has fallen and reduce consumption of those goods whose relativeprice has risen. This dispersion in consumption reduces welfare.

The solution is to prevent price dispersion by stabilizing the price levelby keeping inflation equal to zero.


Woodford versus Friedman

The basic new Keynesian model suggests price stability (i.e., zeroinflation) is optimal.

I Zero inflation eliminates ineffi cient price dispersion.

Milton Friedman argued that a zero nominal rate of interest isoptimal.

I Zero nominal rate eliminates ineffi ciency in money holdings.I Optimal inflation is negative (deflation) at rate equal to real rate ofinterest.

Khan, King, and Wolman (2000) analysis model with both distortionsand conclude optimal inflation is closer to zero than to the Friedmanrule.

Schmidt-Grohe and Uribe (2009), Coibion, Gorodnichenko, andWieland (2012, REStudies): optimal π still small even when ZLBtaking into account.


Basic model —eliminating the steady-state distortion

Assume objective is to minimize

12

ΩEt∞

∑i=0

βi(π2t+i + λx2t+i

).

Note that x∗ has been set equal to zero in loss function

Fiscal subsidy to offset distortion from monopolistic competition.

If x∗ 6= 0, can’t use first order approximations to structural equationsto obtain a correct second order approximation to the representativeagent’s welfare.


Policy Implication of forward-looking models

The basic new Keynesian inflation adjustment equation took the form

πt = βEtπt+1 + κxt ,

where x is output relative to flexible-price output and real marginalcost is (σ+ η)xt .That is, there is no additional disturbance term.

πt = βEtπt+1 + κxt ⇒ πt = κ∞

∑i=0

βiEtxt+i

The absence of a stochastic disturbance implies there is no conflictbetween a policy designed to maintain inflation at zero and a policydesigned to keep the output gap equal to zero.Just set xt+i = 0 for all i ; keeps inflation equal to zero: Blanchardand Galí’s “divine coincidence”.Productivity shocks don’t appear —with only prices sticky, real wageis flexible .


Cost shocks

Assumeπt = βEtπt+1 + κxt + et

where e represents an inflation or cost shock.

Then

πt = κ∞

∑i=0

βiEtxt+i +∞

∑i=0

βiEtet+i

Cannot keep both x and π equal to zero: trade-offs must be made.


Sources of cost shocks

Stochastic wedge between marginal rate of substitution and real wage.

Wedge between flexible-price output and effi cient level of output.I Stochastic markups

Endogenous sourcesI Sticky nominal wages.I Cost channel.I Exchange rate movements, imperfect pass through.


Decompositing outputConsider the following decomposition of output:

Yt =(YtY ft

)(Y ftY pott

)(Y pott

Y et

)Y et

where Y et is the effi cient level of output and Yft is the level of output

under flexible prices and wages, Y pott is potential output (output withconstant markups and flexible prices).In terms of log deviations around the steady state,

yt − y et = xt + x fpott + xpott .

In baseline NK model, x fpott = 0 and xpott is a constant (related tosteady-state markups). So

σ2y−y e = σ2x

and minimizing output around the effi cient level is the same asminimizing xt (Blanchard-Galí ‘divine coincidence’).


Decomposing gaps

In general,σ2y−y e = σ2x + 2σxx fpot + t.i .p.

and minimizing σ2x does not necessarily minimize σ2y−y e . Covariancescan matter.

Optimal policy may not involve minimizing yt − y et since resultingfluctuations in price and wage inflation may be too costly.

I Theory of second best in the face of multiple distortions.


Decomposing gaps

Which fluctuations in yt are due to fluctuations in xt , which due tox fpott , which to xpott , which to y et ?

Which fluctuations in yt are effi cient? Which are ineffi cient?

Chari, Kehoe, and McGrattan (2009) emphasized need to distinguishbetween effi cient and ineffi cient sources of fluctuations.

I May be hard to identify sources of fluctuations.


Gaps and wedges

Effi cient requires thatmrst = mplt

Shocks that cause wedge between these two can beI Exogenous

F Effi cient: preference shifts (χt )F Ineffi cient: shifts in desired markups to wages and prices (µwt , µpt )

I Endogenous

F Ineffi cient: nominal wage and price rigidities (ϕwt , ϕpt ).


The case of flexible prices and wagesFlex-price/wage output

With flexible prices and wages,

ωt = mplt − µpt ; ωt = mrst + µwt

Suppose marginal rate of substitution is

mrst = ηnt + σct + χt

and marginal productivity is

mplt = yt − nt = zt .Then, the steady-state labor equilibrium condition yields

ηnt + σct + χt + µwt = ωt = zt − µpt .

Now using the fact that yt = zt + nt and yt = ct , the flexible-priceequilibrium output y ft can be expressed as

y ft =(1+ η

η + σ

)zt −

(1

η + σ

)(µpt + µwt + χt ) . (3)


Measuring wedges: sticky prices and wages

The wedges ϕwt and ϕpt are defined by

ωt = mplt − µpt − ϕpt

ωt = mrst + µwt + ϕwt

The effi ciency wedge is

mrst −mplt = (ωt − µwt − ϕwt )− (ωt + µt + ϕpt )

= − (µwt + µt )− (ϕwt + ϕpt )


Measuring wedges

The observed wedge:

(ηnt + σct )− (yt − nt ) = − (µt + µwt + χt ) + (ϕwt + ϕpt )

I LHS is observable: RHS isn’t.

With log utility and y = c ,

nt = −(

11+ η

)(µt + µwt + χt − ϕwt − ϕpt )

so employment gives a direct measure of the labor wedge but not itssources.


Effi ciency gaps

Galí, Gertler, and López-Salido (2002) define the “ineffi ciency gap”asthe gap between the household’s marginal rate of substitutionbetween leisure and consumption (mrst) and the marginal product oflabor (mplt).

They divide this gap into the wedge between the real wage and themarginal rate of substitution, which they label the wage markup, andthe wedge between the real wage and the marginal product of labor(the price markup).

mrst −mplt = (mrst −ωt ) + (ωt −mplt )

Based on United States data, they conclude the wage markupaccounts for most of the time series variation in the ineffi ciency gap.

Consistent with the importance of nominal wage rigidity.


Empirical estimates of the wedges: Galí, Gertler, andLopez-Salido (REStat 2007)

Assume σ = η = 1, so lwt = (nt + ct )− (yt − nt ).

1950 1960 1970 1980 1990 2000 20100.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.100 GGLS_PRICEGGLS_WAGE

Figure: Galí, Gertler, and Lopez-Salido’s price gap and wage gap.


Empirical estimates of the wedges: Galí, Gertler, andLopez-Salido (REStat 2007)

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.100

3

2

1

0

1

2

3

4

5G G L S_ G APG G L S_ W AG EG G L S_ PR IC EU N R ATE

Figure: Galí, Gertler, and Lopez-Salido’s effi ciency, wage, and price gaps and USunemployment rate (right axis).


Wedges in the simple exampleThe Justiniano and Giorgio Primiceri (2009) model gap and the price wedge

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 200510

5

0

5

10

15JP model gap based on simple example

JP model gapprice mark up

Figure: The model gap, y − yp and the price wedge from the simple example


Wedges in the simple exampleThe model gap, the price wedge, and the labor wedge

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 200510

5

0

5

10

15

JP model gap based on simple example

JP model gapprice markupwage markup

Figure: The model gap, y − yp , the price wedge, and the wage wedge from thesimple example


Decomposing gaps using DSGE models

Is it a good shock or a bad shock? (Chari, Kehoe, and McGrattan2009)

Suppose there are shocks to the disutility of labor and to the wagemarkup:

mrst =χtN

ηt

C−σt

=(Wt/Pt )

µwt⇒ µwt χtN

ηt

C−σt

=Wt

Pt

What matters is µwt χt —one is a bad shock (µwt ) and one is a good

shock (χt).

Linearized:ηnt + σct − (wt − pt ) = − (χt + µwt )

Can we identify the two shocks?


Explaining the labor wedge

Galí, Smets and Wouter (2011): using unemployment as anobservable (see exercise).

Sala, Söderström, and Trigari (2010): alternatives with markups andpreferences as source of persistence.

Justiniano, Primiceri, and Tambalotti (2012): measurement error andpersistence:

I If wage series (wmt − pt ) measures (wt − pt ) with error, thenηnt + σct = (wmt − pt )− (emt + χt + µwt )

I Use multiple (2) series wm1,t and wm2,t to reduce measurement error.

I Assume χt is AR(1) and µwt is i.i.d.


Gas and wedges in an estimated DSGE modelThe Justiniano, Primiceri and Tambalotti (2012) model gap:


Gas and wedges in an estimated DSGE model

yt =(yt − ypott

)+ ypott


Justiniano, Primiceri and Tambalotti (2012)Using one wage series, wage markups are important.


Justiniano, Primiceri and Tambalotti (2012)Using two wage series, wage markups are small.


Sala, Söderström, and Trigari (2010)


Summary

Gaps/wedges can arise from effi cient shocks and ineffi cient shocks.

Implications for policy are different, so identifing nature of shocks itimportant.

Serious identification issues:I Is the labor wedge due to preference shocks, wage markup shocks, ormeasurement error?


Optimal policy in the basic model

When forward-looking expectations play a role, discretion leads to astabilization bias even though there is no average inflation bias.

Minimize

−12

ΩEt∞

∑i=0


)subject to

πt = βEtπt+1 + κxt + et .

Notice the Euler condition imposes no constraint —use it to solve forit once optimal πt and xt have been determined.

I This would not be case if central bank cares about interest ratevolatility.


DiscretionThe policy problem

Policy maker takes expectations as given — leads to period by periodmaximization.

Problem is to pick πt and xt to minimize

12

(π2t + λx2t

)+ ψt (πt − βπt+1 − κxt − et )

taking Etπt+1 as given.

The first order conditions can be written as

πt + ψt = 0 (4)

λxt − κψt = 0. (5)

Eliminating ψt , λxt + κπt = 0 — this is a targeting rule.


DiscretionBehavior of the interest rate

From the IS equation,

it = Etπt+1 + σ (Etxt+1 − xt ) + rnt .

Using solution,

it =[Aρ− σ

( κ

λ

)(ρ− 1)

]et + rnt = Bet + r

nt .

Shifts in natural rate of interest rn are fully offset.

So optimal policy involves i responding to shocks, but adopting a ruleof the form

it = Bet + rnt

does not ensure a unique rational expectations equilibrium.


Precommitment

When forward-looking expectations play a role, discretion leads to astabilization bias even though there is no average inflation bias.

Under optimal commitment, central bank at time t chooses bothcurrent and expected future values of inflation and the output gap.

Minimize

−12

ΩEt∞

∑i=0


)subject to

πt = βEtπt+1 + κxt + et .

Notice the Euler condition imposes no constraint —use it to solve forit once optimal πt and xt have been determined.


Optimal precommitment

The central bank’s problem is to pick πt+i and xt+i to minimize

Et∞

∑i=0

βi[12

(π2t+i + λx2t+i

)+ ψt+i (πt+i − βπt+i+1 − κxt+i − et+i )

].

The first order conditions can be written as

πt + ψt = 0 (6)

Et(πt+i + ψt+i − ψt+i−1

)= 0 i ≥ 1 (7)

Et(λxt+i − κψt+i

)= 0 i ≥ 0. (8)

Dynamic inconsistency —at time t, the central bank sets πt = −ψtand promises to set πt+1 = −

(Etψt+1 − ψt

). When t + 1 arrives, a

central bank that reoptimizes will again obtains πt+1 = −ψt+1 —thefirst order condition (6) updated to t + 1 will reappear.


Timeless precommitment

An alternative definition of an optimal precommitment policy requiresthe central bank to implement conditions (7) and (8) for all periods,including the current period so that

πt+i + ψt+i − ψt+i−1 = 0 i ≥ 0

λxt+i − κψt+i = 0 i ≥ 0.Woodford (1999) has labeled this the “timeless perspective”approachto precommitment.


Timeless precommitment

Under the timeless perspective optimal commitment policy, inflationand the output gap satisfy

πt+i = −(

λ

κ

)(xt+i − xt+i−1) (9)

for all i ≥ 0.Woodford (1999) has stressed that, even if ρ = 0, so that there is nonatural source of persistence in the model itself, a > 0 and theprecommitment policy introduces inertia into the output gap andinflation processes.

This commitment to inertia implies that the central bank’s actions atdate t allow it to influence expected future inflation. Doing so leadsto a better trade-off between gap and inflation variability than wouldarise if policy did not react to the lagged gap.


Illustrating commitment versus discretion in the simple NKmodel

2 4 6 8 102

1.5

1

0.5

0Output Gap

2 4 6 8 100.2

0

0.2

0.4

0.6

0.8Inflation

2 4 6 8 100

0.5

1

1.5

2Nominal interest rate

2 4 6 8 100.2

0

0.2

0.4

0.6

0.8

1

1.2Cost Shock

Taylor RuleFull Commitment Optimal PolicyOptimal Discretion

MATLAB\dynare\NKmodels\Gerzensee2012\NKM_basic\Graphs_Compare.mCarl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 63 / 319

Improved trade-off under commitment

The difference in the stabilization response under commitment anddiscretion is the stabilization bias due to discretion.

Consider a positive inflation shock, e > 0.

A given change in current inflation can be achieved with a smaller fallin x if expected future inflation can be reduced:


Requires a commitment to future deflation.

By keeping output below potential (a negative output gap) for severalperiods into the future after a positive cost shock, the central bank isable to lower expectations of future inflation. A fall in Etπt+1 at thetime of the positive inflation shock improves the trade-off betweeninflation and output gap stabilization faced by the central bank.


The case of a persistent cost shock

2 4 6 8 103

2.5

2

1.5

1

0.5

0Output Gap

2 4 6 8 100.2

0

0.2

0.4

0.6

0.8

1

1.2Inflation

2 4 6 8 100.5

0

0.5

1

1.5Nominal interest rate

2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4Cost Shock

Taylor RuleFull Commitment Optimal PolicyOptimal Discretion


Illustrating policy effects in S-W model: inflation shock

5 10 15 20 25 30 35 400.5

0.4

0.3

0.2

0.1

0Output gap

5 10 15 20 25 30 35 400.5

0.4

0.3

0.2

0.1

0Actual output

5 10 15 20 25 30 35 400.1

0

0.1

0.2

0.3

0.4Inflation

5 10 15 20 25 30 35 400.1

0

0.1

0.2

0.3

0.4

0.5

0.6Nominal interest rate

Estimated Taylor RuleOptimal Commitment PolicySimple Taylor rule


Uncertainty

Standard analysis with additive errors leads to certainty equivalence(linear structural equations, quadratic objective function)

I Optimal policy depends on future forecasts but not the uncertaintysurrounding those forecasts.

Suppose the true model of the economy is given by

yt+1 = A1yt + A2yt/t + Bit + ut+1, (10)

where yt is a vector of macroeconomic variables (the state vector),yt/t is the optimal, current estimate of yt/t , and it is the policymaker’s control instrument.

ut+1 represents a vector of additive, exogenous stochasticdisturbances, assumed equal to Cet+1 where the vector e is a set ofmutually and serially uncorrelated disturbances with unit variances.

A1, A2, and B are matrices of the model parameters.


Sources of model specification error

Suppose the policy maker’s estimates of A1, A2, and B are denotedA1, A2, and B, while yt/t denotes the policy maker’s estimate of thethe current state yt .

Then, letting A = A1 + A2 and A = (A1 + A2), we can write thepolicy maker’s perceived model in the form

yt+1 = Ayt/t + Bit + C (et+1 + wt+1) (11)

where

wt+1 = C−1 [(A− A) yt/t + (B − B) it + (C − C ) et+1]+C−1A1 (yt − yt/t ) + C−1A (yt/t − yt/t ) . (12)


Sources of model specification error

wt+1 = C−1 [(A− A) yt/t + (B − B) it + (C − C ) et+1]+C−1A1 (yt − yt/t ) + C−1A (yt/t − yt/t ) .

1 Model mis-specification: errors that arise if the policy maker’sestimate of the parameters of the model differs from their true values.This term also captures errors in modelling the structural impacts ofexogenous disturbances.

2 Imperfect information: errors the policy maker incurs in estimatingthe current state of the economy.

3 Asymmetric and/or ineffi cient forecasting: informationalasymmetries such as occur when the private sector has differentinformation than the policy maker does.


Multiplicative uncertainty: Brainard (1967)

Model (y is the output gap, π is inflation):

πt = βEtπt+1 + κtxt + et

Assume κt , is stochastic, κt = κ + vt , where vt is a white noiseprocess.

Under discretion, the policy maker takes Etyt+1 and Etπt+1 as given.The first order condition yields

xt = −(

κ

λ+ σ2v

)et

Reaction with more caution.

Does Brainard’s result generalize? No. Craine (1979), Söderström(2002).


Robust control: The reference modelPolicy maker has reference model of form

yt+1 = Ayt + Bit + C et+1

True model is

yt+1 = Ayt + Bit + C (et+1 + wt+1) , (13)

I In the robust control literature, wt+1 represents unknown specificationerrors.

I wt+1 is not simply an exogenous disturbance like et+1 but may dependon the history of yt .

The policy maker views Ayt/t + But + C et+1, i.e., the case wt+1 = 0,as a good “approximating model” to the true but unknown model.It is a good approximation in the sense that

∞

∑i=0

βiw ′t+iwt+i ≤ η0, (14)


The intuition

Strategic game involving the policy maker and an evil agent whoattempts to make life hard for the policy maker.

Leads to a min-max strategy by the policy maker, with the policyinstrument chosen to minimize the worst-case outcome.

Equilibrium is given by the solution to

minwmaxu

Et

[∞

∑t=0

βt−r(yt , ut ) + θβw ′t+1wt+1

]

where yt is the state, ut is the policy maker’s control, r (yt , ut ) is thequadratic loss, and β is the discount factor.

As θ → ∞, evil agent is more constrained. Standard case whenθ = ∞.


Using a distorted model

The policy maker replaces the model of the economy with a distortedmodel, one that incorporates the worst case process for wt+1.

The evil agent sets wt+1 to maximize the policy maker’s loss function.I The value of wt+1 for which the worst-case outcome occurs can beexpressed as a function of the state vector, Kyt .

Substituting wt+1 = Kyt into (13) yields the distorted model:

yt+1 = (A+ CK )yt + Bit + C et+1. (15)

The policy maker now treats this distorted model as the true model ofthe economy and minimizes loss subject to (15).

I Once the policy maker has substituted in Kyt for wt+1, the policyproblem is reduced to a standard one —certainty equivalence holds.


Using a distorted model

A Bayesian policy maker, faced with model uncertainty, assigns aprobability to each possible model, where these probabilities reflectthe policy maker’s assessment of the likelihood of each model.

A policy maker concerned with robustness bases policy on a distortedmodel but then proceeds to act as if there were no longer any modeluncertainty.

An example: Does it pay to underestimate inflation persistence orover estimate it? Yes.

I Coenen (2003), Angeloni, Coenen, and Smets (2003), and Walsh(2003).


Preference for robustness

The same policy that results from a policy maker employing thedistorted model is also obtained when the policy maker believes thetrue model is given by

yt+1 = Ayt + Bit + C et+1,

and she maximizes an objective function that contains an additionaladjustment for risk.

Specifically, the policy maker’s preferences incorporate an additionalsensitivity to risk. Similar risk sensitive preferences have been studiedby Epstein and Zin (1989) and Weil (1990).

Preferences are of the form

Vt = U [ct ,Vt+1]

Allows risk aversion and elasticity of intertemporal substitution to beseparated.


Comparing the standard optimal rules and robust control

Both approaches lead to same instrument or targeting rule.

Consider simple example —new Keynesian model with ψt theLangrangian multiplier on the NKPC.

Standard approach —first order conditions:

πt + ψt − ψt−1 = 0

andλxt − κψt = 0.

Combine to yield robustly optimal targeting rule:

πt = −(

λ

κ

)(xt − xt−1) .


Comparing the two approachesEquilibrium under the robustly optimal rule

Together with the inflation adjustment equation, this yields[1+ β+

(κ2

λ

)]xt = βEtxt+1 + xt−1 −

( κ

λ

)et , (16)

which can be jointly solved with the process for et given by

et = ρeet−1 + εt . (17)

under rational expectations.


Robust control

Robust control problem is

minx ,π

maxw ,e

Erct∞

∑i=0

βi[(12

)π2t+i +

(12

)λxx2t+i

−(12

)βθw2t+1+i

+ψt+i (πt+i − βπt+i+1 − κxt+i − et+i )+ϕt+i (ρeet+i + εt+i+1 + wt+i+1 − et+i+1) .


Robust control

The policy maker’s first order conditions include

π + ψt − ψt−1 = 0, (18)

λxt − κψt = 0, (19)

The evil agent’s first order conditions include

−ϕt + ρe ϕt −(1β

)ϕt−1 = 0, (20)

and−θwt+1 + ϕt = 0. (21)


Robust controlEquilibrium

Combining first two with the inflation adjustment equation yields[1+ β+

(κ2

λ

)]xt = βErct xt+1 + xt−1 −

( κ

λ

)et , (22)

which is identical to equation obtained in standard case except for theformation of expectations.Evil agent’s first order conditions imply that

ϕt−1 = βρe ϕt − βψt = βρe ϕt − β

(λ

κ

)xt .

Advancing this expression one period, taking expectations, solving theresulting expression forward implies that

wt+1 = −(

βλ

κθ

) ∞

∑i=0(βρe )

iErct xt+1+i . (23)


Extensions: A simple rule for the evil agent

Suppose the evil agent commits to a contingency rule that makesmisspecification a function of exogenous state variables.

Specifically, assume evil agent commits to wt+1 = Ket .

Define ρe = ρe +K . Then, for any choice of K such that | ρe |< 1,the policy maker’s problem becomes

miniE rct

∞

∑i=0

βi(

12

)π2t+i +

(12

)λxx2t+i −

(12

)βθ(Ket+i )2

+ψt+i (πt+i − βπt+i+1 − κxt+i − et+i )+ϕt+i (ρeet+i + εt+i+1 − et+i+1)

.


Extensions: A simple rule for the evil agent

This is a standard problem with the shock process replaced by thedistorted process

et+1 = ρeet + εt+1. (24)

The policy maker (and the public) takes ρe (i.e., K ) as given.

The optimal targeting criterion is xt = xt−1 − (κ/λ)πt , which isindependent of ρe (and therefore K ).

This independence reflects the fact that the standard targetingcriterion is designed to be robust with respect to exactly the type ofmodel mis-specification of the disturbance process that is reflected in(24). (see Giannoni and Woodford 2003).


Using multiple models

McCallum has long argued for evaluating simple rules in multiplemodels.

Levin and Williams (2003).

Forward-looking models too easy to control — rules optimal inforward-looking models tend to do poorly in backward-looking models.

Standard approach —use different models, but outcomes areevaluated using a fixed loss function.

I What if loss function for model A is different from loss function formodel B?

I Theory says this will be the case.

Example: greater nominal rigidity reduces output elasticity ofinflation.

I requires greater output gap variability to control inflation;I but theory says weight on gap fluctuations in loss function should fall.


Structural inflation inertiaImportant for relative performance of different targeting rules (Walsh2003).Suppose

πt − γπt−1 = β (Etπt+1 − γπt ) + κxt + et .

Woodford (2003) shows that second order approximation to welfare isproportional to

−(12

) ∞

∑i=0

[(πt − γπt−1)

2 + λxx2t].

Define zt ≡ πt − γπt−1. Then model becomes

min(12

) ∞

∑i=0

[(πt − γπt−1)

2 + λxx2t]

subject tozt = βEtzt+1 + κxt + et .

Loss independent of γ!Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 84 / 319

Is the ZLB a constraint?

D istr ib utio n o f Federal Fund s R ate: Jan 1960 Mar . 2012

5 0 5 10 15 20 250.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

Figure: Distribution of the federal funds rate, Jan. 1960 - Mar. 2012


The zero lower bound

CausesI Non-fundamentals-based liquidity traps

F Expectationally driven

I Fundamentals-based liquidity traps (will address later)

F Negative shock to the equilibrium (Wicksellian) real rate of interest

Potential constraint on monetary policy


The Taylor principle and liquidity trapsNon-fundamentals-based liquidity traps

Benhabib, Schmit-Grohé, and Uribe (2001, 2002) have argued thatdeflationary paths cannot be ruled out if policy satisfies the TaylorPrinciple.

I Multiple steady-state equilibria in forward-looking expectational models.I The argument is based on the observation that the nominal rate ofinterest cannot fall below zero.

Explosive deflations would eventually force the nominal interest rateto zero, but the nominal rate is then prevented from falling further.

They argue that simple and seemingly reasonable monetary policyrules that follow the Taylor Principle in changing the nominal interestrate more than one-for-one is response to changes in inflation canintroduce the possibility the economy will be caught in a deflationaryliquidity trap.


The Taylor principle and liquidity trapsSuppose we have a model displaying superneutrality. The “monetary”side of the model is summarized by

it = rnt + Etπt+1

it = g(πt )

where rnt is exogenous with respect to inflation and the nominalinterest rate and g(π) is the policy rule.For simplicity, let

g(πt ) = rnt + π∗ + δ (πt − π∗)

= rnt + (1− δ)π∗ + δπt

Then combining these equations,

Etπt+1 = (1− δ)π∗ + δπt

There is a unique stationary equilibrium inflation rate equal to π∗.However, if the time t inflation rate is not equal to π∗, the inflationprocess is unstable.


The Taylor principle and liquidity trapsA simple example

π(t)

π(t+1)

π∗

π∗∗

Figure: Liquidity trap caseCarl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 89 / 319

The zero lower boundFundamentals-based liquidity traps

Optimal policy an a basic new Keynesian model implies the policyinterest rate moves one-for-one with the equilibrium real (natural,Wicksellian) interest rate:

it = rnt + π∗ + δ (πt − π∗) ≥ 0.

Negative shock to rnt could require it to be negative —ZLB preventsthis.

Deflationary trap:

rt = it − Etπt+1 = −Etπt+1


Policy at the ZLBIf i = 0, short-term government debt and money perfect substitutes.

I Altering B +M doesn’t causes returns to adjust.

Consider the nominal budget constraint of a household holding moneyplus short-term government debt, denoted B yielding nominal returnib :

PtYt + (1+ ibt )Bt +Mt ≥ PtCt + PtTt + Bt+1 +Mt+1.

The budget constraint can be expressed in real terms as

Yt +(1+ rbt

)Ft ≥ Ct + Tt +

(ibt

1+ πt

)mt + Ft+1,

where Ft ≡ bt +mt equals real holds of financial assets.If the nominal interest is zero, 1+ rb = 1/(1+ π),

Yt +(

11+ πt

)Ft ≥ Ct + Tt + Ft+1,

and m drops out.Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 91 / 319

Is the ZLB a constraint?

Is the ZLB a constraint on monetary policy?

Conventional model based on an expectational IS relationship:

xt = Etxt+1 −(1σ

)(it − Etπt+1 − rnt )

Interest rates —both current and expected future matter:

xt = −(1σ

)(it − Etπt+1)−

(1σ

)Et

∞

∑i=1(it+i − πt+1+i )

+

(1σ

)Et

∞

∑i=0rnt+i ,

Reflects a narrow view of the transmission mechanism —no role forquantitative easing or credit easing policies.


Conventional instruments at the ZLB

Even at the ZLB, policy has the potential to influence real spending ifit can affect expectations of future real interest rates.

I Eggertsson and Woodford (2003)

If it = 0 and is expected to remain at zero until t + T , then

xt =

(1σ

) T

∑i=0

Etπt+1+i −(1σ

)Et

∞

∑i=T+1

(it+i − πt+1+i )

+

(1σ

)Et

∞

∑i=0rnt+i .

Raising expected future inflation or committing to lower futurenominal rates can stimulate current spending.

I Cost of ZLB low in linear models when central bank is credible (Nakov2008)


Promising future inflation

Optimal policy at the ZLB involves promising future inflation. This isdone by keeping interest rates low even when the ZLB no longer binds(Eggertsson and Woodford 2003).

Central banks have been reluctant to promise higher future inflation

Contrasts with recommendations made to the Bank of Japan:I Krugman (1998), McCallum (2000), Svensson (2001, 2003), andAuerbach and Obstfeld (2005)

I Bernanke (2000) versus Bernanke (2010).

Communicating clearly the conditional nature of future interest ratepaths cited as concern.

Commitment requires promises be fulfilled —have to deliver higherfuture inflation.

Central banks may lack the credibility to steer future expectations(Bodenstein, Hebden, and Nunes 2010).


Promising future inflationFour-period example of Bodenstein, Hebden and Nunes (2010)

Suppose economy is described by

xt = Etxt+1 −(1σ

)(it − Etπt+1 − rnt )

πt = βEtπt+1 + κxt

Policy objective is to minimize

12

Et∞

∑i=0


)In period 1, the economy is at the zero lower bound: rn1 < 0 andi1 = 0.In periods 2, 3 and 4, the economy is out of the ZLB so thatxt+i = πt+i = 0 for i ≥ 2 is feasible. Assume that πt+4 = xt+4 = 0.(This is what makes this a simple example.)The issue is what happens in periods 2 and 3 and how this affects theoutput gap and inflation in period 1.


Promising future inflation: examplePath of interest rate relative to equilibrium real rate under discretion and full commitment

1 1.5 2 2.5 3 3.5 43

2

1

0

1

2

3

period

perc

ent

discretioncommitment: µ = 1

Figure: The nominal rate under discretion and full commitment when ZLB isbinding only in period 1 (figure shows i − rn)


Promising future inflation: examplePath of inflation and the output gap under discretion and full commitment

1 1.5 2 2.5 3 3.5 41

0.5

0

0.5

1Inflation rate

period

perc

ent

discretionhigh credibil ity:µ = 1

1 1.5 2 2.5 3 3.5 44

2

0

2

4Output gap

period

perc

ent

Figure: Inflation and the output gap under discretion and full commitment whenthe ZLB is binding only in period 1


Promising future inflation: examplePath of inflation and output gap with imperfect credibility

1 1.5 2 2.5 3 3.5 41

0.5

0

0.5

1Inflation rate

period

perc

ent

discretionhigh credibil ity: µ = 1low credibility: µ = 0.25

1 1.5 2 2.5 3 3.5 44

2

0

2

4Output gap

period

perc

ent


Promising future inflation: examplePath of interest rate relative to equilibrium real rate with imperfect credibility

1 1.5 2 2.5 3 3.5 43

2

1

0

1

2

3

period

perc

ent

discretionhigh credibil ity: µ = 1low credibility: µ = 0.25

Figure: The nominal rate under discretion and full commitment when ZLB isbinding only in period 1 (figure shows i − rn)


Promising future inflation: example

A lack of credibility makes it harder to stabilize in the face of the ZLB;

With less credibility, future promises must be more extreme;

“Promising low interest rates for an extended period of time”may bea sign of a lack of credibility;

Promising low interest rates in the future and also promising noinflation is an inconsistent policy.

When current credibility is low, the central bank has to promise lowerfuture rates, but that means the cost of fulfill promises is higher,raising the incentive to deviate.

I Central banks with low credibility face the greatest temptation to breaktheir promises.


Raising the inflation target

Raising the inflation target raises average nominal interest rates andmakes hitting the ZLB less likely.

Trade-off —steady-state loss of higher inflation against ability toimprove stabilization.

Schmidt-Grohe and Uribe (2009), Coibion, Gorodnichenko, andWieland (2012, REStudies): optimal π still small when ZLB takinginto account.


Reforming IT: Using expectations as automatic stabilizersunder PLT


Price level targeting —Svensson (1999), Vestin (2006), Walsh (2003),Cateau, et. al (2008), Dib, et. al. (2008), Kryvstov, et. al. (2008).Central banks may find it easier to commit to objectives than tofuture policy actions.

I Distorting objectives in a discretionary environment can improveoutcomes (Walsh 1995)

Outcomes to shocks under discretionPLT relative to IT

Disturbance Price-level targetingInflation shocks Potentially BetterDemand shocks, no ZLB SameDemand shocks, ZLB Better


Price level targeting and the ZLB

Optimal policy at the ZLB can be implemented via a time-varyingprice-level target (Eggertsson and Woodford 2003).

Intuition —under commitment, central bank has many tools even ifcurrent policy rate at zero.

I Can promise future low interest rates and a boom.I This generates expectations of future inflation which raises currentinflation and lowers current real interest rate.

Promise to generate inflation to achieve price-level target.


Price level targeting

Vestin (2006) shows price level targeting can replicate the timelessprecommitment solution if the central bank is assigned the lossfunction p2t + λPLx2t in an environment of discretion.

Under timeless precommitment,

πt = (1− L)pt =(

λ

κ

)(xt − xt−1)⇒ pt =

(λ

κ

)xt

Price level targeting makes inflation expectations act as an automaticstabilizer.

Walsh (2003) adds lagged inflation to the inflation adjustmentequation and shows that the advantages of price level targeting overinflation targeting decline as the weight on lagged inflation increases..


PLT: other considerations

Advantages of PLT require that expectations act as automaticstabilizers.

I Raises issues of credibility and learning

Switching policy regimes in a crisis risks gains in credibility achievedby inflation targeters.

I When adopted, the choice of price index, the underlying trend inflationrate, and the speed with which deviations from target path areexpected to be reversed are all important.

Walsh (2003) adds lagged inflation to the inflation adjustmentequation and shows that the advantages of price level targeting overinflation targeting decline as the weight on lagged inflation increases..


The zero lower bound: other solutions for getting out

Svensson’s “foolproof way”I Depreciation as visible means of commiting to a higher price level

Fiscal policy: “fiscal policy must be seen not to be commited to...conventional prescriptions for good fiscal policy...”. (Sims 2000, p.969, italics in original) —more on this later.

Quantitative easing/credit easing policies —more on this later.


Fed’s balance sheet: Conventional

Jan2007 Jan2008 Jan2009 Jan2010 Jan2011 Jan20120

500

1000

1500

2000

2500

3000

billi

ons

of $

Traditional Security HoldingsLending to Financial InstitutionsLiquidity to Key Credit Markets


Fed’s balance sheet: Conventional and unconventional

Jan2005 Jan2008 Jan2009 Jan2010 Jan2011 Jan20120

500

1000

1500

2000

2500

3000

billi

ons

of $

Traditional S ecurity H oldingsLending to Financial InstitutionsLiquidity to K ey Credit MarketsLT Treasury P urchasesMortgaged B acked S ec. P urchases


Unconventional policies: Quantitative easing

So far, monetary policy only works via the Wicksellian interest rategap (it − Etπt+1)− rnt .

I Pre-crisis consensus: No direct role for money —M not a separatepolicy tool once i set

At the ZLB —money and short-term assets perfect substitutes

Quantity of money can still matterI Depends on the properties of money demandI Depends on monetary expansion being permanent

Paying interest on reserves gives central bank two instruments:I Policy interest rate and rate on reservesI Policy interest rate and monetary aggregate.


Money in the NK model

Assume money generates utility directly.

Demand for money determined by equating marginal rate ofsubstitution between real money balances and the opportunity cost ofholding money.

I The opportunity cost depends on the nominal rate of interest.

In linearized form:mt − pt = γyt − ηit .

Given it and yt determined by rest of system, this equation justdetermines mt . No separate role for money.

I Assumes separable utility.


Money in the NK model

With interest paid on money, then m and i (or i and im) becomeseparate instruments.

In linearized form:

mt − pt = γyt − η (it − imt ) .

Given it and yt determined by rest of system, this equation no longerdetermines mt unless imt is also specified.

Set im = i to eliminate the Friedman distortion without requiring thati = 0.

Issue is whether the quantity of money matters.


Does money matter at the ZLB?

Consider a standard (linearized) money demand equation:

mt − pt = γyt − η (rt + Etπt+1) = γyt − η (rt + Etpt+1 + pt ) .

Solving forward:

pt = Et∞

∑j=0

(η

1+ η

)(mt+j − γyt+j + ηrt+j ) .

So the future path of money matters for prices today.


Unconventional policies: Affecting long-term interest ratesStandard models of the term structure imply the m-periodzero-coupon bond rate is related to the expected future path ofshort-term real rate according to

rmt =(

1m− 1

)Et

m−1∑j=0

(it+i − πt+i ) +Ψm,t , (25)

where Ψm,t is a risk premium.When Ψm,t is viewed as exogenous, (25) implies that only byinfluencing expectations about future interest rates and inflation canthe central bank affect the real economy at the ZLB.If Ψm,t is endogenous and varies with factors influenced by monetarypolicy, then altering the path of the future policy rate may not be thesole means of affecting the economy.Question: are there other transmission channels besides the expectedfuture path of interest rates that might provide the means to affectlong-term rates? (Particularly relevant at the ZLB.)


Modigliani-Miller and open market operations

Assume all agents can freely buy and sell assets and assets are valuedonly because of their payouts.

Then price of asset j with payoffs xj (s) in future state s is

pj =S

∑s=1

π(s)m(s)xj (s) =S

∑s=1

π(s)βUc [c(s)]Uc (ct )

xj (s)

= βEt

(Uc [c(s)]Uc (ct )

)xj (s),

where π(s) is the probability of state s.

Asset quantities do not appear.

Money is different —usually viewed as having a non-pecuniary return.


Modigliani-Miller

Wallace (AER 1981) demonstrated a Modigliani-Miller result for openmarket operations.

Asset prices are independent of the central bank’s balance sheet —open market operations and the form they take (short-term gov’tdebt, long-term gov’t debt, private assets) are irrelevant.

Conditions needed (Curdia and Woodford JME 2011):I If assets are valued only for the pecuniary returns andI If all investors can purchase arbitrary quantities of the same assets atthe same prices.

Government purchases of assets doesn’t alter total risk.


Budget identities

Suppose household can hold one-period bonds, two-period bonds, andbase money. The budget constraint in nominal form would take the form

PtYt +DP1,t−1 + p1,tDP2,t−1 +Mt−1 ≥ PtCt + PtTt + p1,tDPt

+p2,tDP2,t +Mt .

here p1,t is the price of a one period discount bond (so p1,t = 1/(1+ i1,t )and p2,t is the price of a two period bond. Now define financial wealth as

Ft ≡ DP1,t−1 + p1,tDP2,t−1 +Mt−1


Budget identities

One can rewrite the budget constraint as

PtYt + Ft = PtCt + PtTt + p1,t(DP1,t + Etp1,t+1DP2,t +Mt

)+ (p2,t − p1,tEtp1,t+1)DP2,t + (1− p1,t )Mt

= PtCt + PtTt + p1,tFt+1+ (p2,t − p1,tEtp1,t+1)DP2,t + (1− p1,t )Mt

The one period interest rate is

1+ i1,t ≡1p1,t

. (26)

The two period rate is

(1+ i2,t )2 ≡ 1

p2,t


No arbitrage conditions

Under the expectations hypothesis,

p2,t = p1,tEtp1,t+1 (27)

To first order,

i2,t =(12

)(i1,t + Et i1,t+1)

So the budget constraint reduces to

PtYt + Ft = PtCt + PtTt + p1,tFt+1 + (1− p1,t )Mt .

At the ZLB, p1,t = 1 and the budget constraint becomes

PtYt + Ft = PtCt + PtTt + p1,tFt+1.

The composition of the portfolio between money, one-period binds ortwo-period bonds is irrelevant.


Portfolio balance models

Assumes imperfect asset substitutability, asset market segmentation

Monetarists vs. Keynesians debatesI Meltzer 1995, Tobin 1969;I Debate about quantitative significanceI Andrés, López-Salido, and Nelson (2004), Goodfriend (2000, 2010).

Empirical evidenceI Modigliani and Sutch (1967): Found little evidence that OperationTwist mattered in the 1960s;

I Clouse, et. al. (2003). Bernanke, Reinhart, and Sack (2004): It wouldrequire extremely large open market operation in non-standard assetsto have a significant impact on yields;

I Gagnon et al (2011), Joyce, Lasaosa, Stevens, and Tong (2011).


Curdia and Woodford (2011)“The central-bank balance sheet as an instrument of monetary policy,”, Journal ofMonetary Economics 58(1), Jan. 2011, 54-79.

Wallace irrelevance result:I Open market operations are neutral if (i) assets valued only for theirpecuniary returns (ii) all investors can purchase arbitrary quantities ofthe assets at the same market prices.

Under these conditions, size and composition of the central bank’sbalance sheet are irrelevant for the equilibrium.

Monetary policy can still matter by affecting the nominal interest rateon overnight balances.


The Curdia-Woodford model: households

Household preferences:

maxE0∞

∑t=0

βt[uτt (i ) (ct (i) : ξt )−

∫ 1

0v τt (i ) (ht (j : i) : ξt )

]where τt (i) ∈ [b, s ] is the household’s type and for τ = b, s,

uτ (c : ξ) =c1−

1στ (C τ)

1στ

1− 1στ

and C is an exogenous preference disturbance.

ct (i) is a Dixit-Stiglitz aggregator with elasticity θ.



Household supplies continuum of labor types j with disutility of workgiven by

v τ (h : ξt ) =ψt1+ v

h1+v H−v

and H is an exogenous disturbance.

Production of good j but a monopolisticlly competitive supplier:

yt (j) = Atht (j)1/φ, φ > 1



Each period, with probability 1− δ, new household type is drawn (sowith probability δ household type remains same as in pervious period).

I When new type drawn, it is b with probability πb and s withprobability πs = 1− πb .

I Assumeubc > u

sc for all c

I So a type b is more impatient to consume.I With heterogeneity among households, there is a role for financialintermediation. Type b wants to borrow; type s wants to save.

Portfolio options available to households —deposits at or loans frombank and government bonds.

I Bonds perfect substitute for deposits so both yield the same return.I Forcing lending to go through banks is a financial frictions.



Households can sign state contingent contracts to insure againstaggregate risk and risk of type, but receives transfers from theseinsurance contracts only occasionally.

I Implies that all households have same expectations about marginalutility of consumption far enough into the future.

Consumption is the same for all households of same type, and is afunction of λτ

t , time t marginal utility of type τ = b, s.

Two Euler conditions:

λτt = βEt

(1+ iτt1+ πt+1

)[δ+ (1− δ)πτ] λ

τt+1 + (1− δ)(1− πτ)λ

−τt

where λ−τ is marginal utility of type not τ.


Inflation

Calvo price adjustment but driving force for inflation now depends onboth λb and λs .

I Can express new Keynesian Phillips curve as a standard linearizedequation with the addition of

Ωt = λbt − λst ,

the marginal utilities gap. (see Curdia and Woodford 2009).I This in turn will be related to the credit spread.I Leads to a type of cost channel as in Christiano, Eichenbaum, andEvans (2005) or Ravenna and Walsh (2006).


Financial intermediariesCredit spread is

1+ωt ≡1+ ibt1+ idt

≥ 0

where id is also the rate on government debt.Intermediaries take in deposits and make one-period loans. They alsohold reserves Mt with the central bank that pay imt .Intermediaries take interest rates as given. The loan rate exceeds thedeposit rate for two reasons:

I Curdia and Woodford assume resources need to be used in originatingloans.

I Some borrowers do not repay.

Intermediary balance sheet:

dt = mt + Lt + χt (Lt ) + ΞPt (Lt ,mt ) + πFIt

where m are real reserve holdings and L are loans, χt (Lt ) is thevolume of bad loans extended, ΞPt (Lt ,mt ) are real resource costs andπFIt are payouts to shareholders.


Financial intermediaries

Payout on deposits:(1+ idt

)dt = (1+ ibt )Lt + (1+ i

mt )mt .

Combining,

πFIt = dt −mt − Lt − χt (Lt )− ΞPt (Lt ,mt )

=(1+ ibt )Lt + (1+ i

mt )mt

1+ idt−mt − Lt − χt (Lt )− ΞPt (Lt ,mt )

or

πFIt =

(idt − ibt1+ ibt

)dt −

(imt − ibt1+ ibt

)mt + χt (Lt ) + ΞPt (Lt ,mt )


Financial intermediaries

This last equation implies the FOC for loans is

χLt (Lt ) + ΞPLt (Lt ,mt ) =(ibt − idt1+ idt

)= ωt (28)

The FOC for m is

−ΞPmt (Lt ,mt ) = δmt ≡idt − imt1+ idt

(29)

Market clearing in the loan market:

bt = Lt + Lcbt

where Lcb represents private sector borrowing from the central bank.


Central bank policy

Balance sheet:mt = Lcbt + b

cbt

where bcbt is central bank holdings of government debt.

Resource costs of central bank lending to private sector is Ξcbt (Lcbt ).Central bank pays interest imt on reserves and receives idt on itsholdings of government debt.

Demand for reserves and spread δm satisfy joint inequalities:

mt ≤ mdt (Lt , δmt )

δmt ≥ 0.

I At least one must hold with equality.I If mt = mdt , then δm > 0 and idt > i

mt .


Central bank policy

id is the policy rate.I By adjusting m, central bank can control δm .I By adjusting imt , it can control level of i

dt for given δmt :

0 ≤ imt ≤ idt .

Three dimensions of policy: Mt , imt and composition of balance sheetbetween Lcb and bcb .

I Or interest rate idt , Mt , (which then determines imt ) and credit policy

Lcbt .


Welfare

Depends on the standard new Keynesian variables plusI Ω —an increase in the spread between marginal utilities of twohousehold types reduces welfare because it is associated with a lesseffi cient allocation since a social planner would equate marginal utilityacross the two agents.

I Ξ since this is a resource cost of financial intermediation.

Friedman rule holds — supply reserves up to their satiation level m.I Note that this does not require id = 0, only δm = 0.


Optimal policy: quantitative easing

If m is the satiation level of m, then optimal money supply policy hasm = m but there is no value in expanding m beyond this point except

I since Lcbt ≤ mt , it could make sense (i.e., be optimal) to expand mabove m if it were necessary to increase lending by the central bank tothe private sector and the central bank had already reduced its holdingsof government debt (recall m = Lcb + bcb).

In the Curdia-Woodford model, there is no benefit directly ofexpanding m above m in general — i.e., no role for quantitative easingif the increase in reserves finances central bank purchases ofgovernment debt (not additional lending to the private sector).


Optimal policy: quantitative easing

This irrelevance result is conditional on the policy paths for idt , imt ,

and Lcbt .I From central bank’s balance sheet,

mt − bgt = Lcbt

so if Lcbt is given, the irrelevance result involves open market operationsin government debt and when ιd = im reflects a classic liquidity trap.

I Auerbach and Obstfeld (2005) argue a permanent increase in M iseffective even when id is currently at the zero bound, but that isbecause such an increase would not be consistent with the policy rulefor id .

I Therefore, Auerbach and Obstfeld are considering a rise in M and achange in the expected future path of the policy rate.


Optimal policy: credit policies

Assume m = m. Credit policy involves the central bank sellinggovernment bonds and making loans to the private sector.

How does Lcb affect welfare?I For given b (private sector borrowing) it reduces L (private lending)and therefore saves on the resource cost of private intermediation.

F This gain only occurs because Curdia and Woodford assume the centralbank can lend costlessly — i.e., the central bank has an advantage inlending to the private sector over private intermediaries. Very unlikelyso they add a resource cost Ξcb(Lcbt ) for central bank lending.

I Since from (28)χLt (Lt ) + ΞPLt (Lt ,mt ) = ωt ,

a fall in L reduces ω, the spread between ib and id falls and thisimproves welfare.



Assume Ξcb(0) > 0 then if Ξcb(0) is suffi ciently high, a Treasuriesonly policy will always be optimal. However, there can be a criticalΞcb,crit such that if it exceeds Ξcb(0), credit policies optimally comeinto play.

They show this is more likely to be the case at the zero lower bound,especially if the central bank cannot commit to a future path for thepolicy rate.



Optimal first order conditions for Lcbt takes for

ϕΞ,t

[ΞpLt

(bt − Lcbt

)− ΞcbL

(Lcbt)]+ ϕω,t

[Ξp′′t

(bt − Lcbt

)+ χ′′t

(bt − Lcbt

)]≤ 0

Lcbt ≥ 0where ϕΞ,t is shadow value of a reduction in intermediation costs andϕω,t is shadow value of a reduction in the spread ω where over barindicates these are evaluated when id = im = 0 so m = m.

Results depend on nature of the financial shock.I Shocks to private sector intermediation costs ΞPt (Lt ,mt ).I Shocks to loan defaults χt (Lt ).



Define

Ξcb,critL = ΞpL(bt − Lcbt

)+

ϕω,t

ϕΞ,t

[Ξp′′t

(bt − Lcbt

)+ χ′′t

(bt − Lcbt

)].

Then ifΞcbL

(Lcbt)< Ξcb,critL

optimal policy will set Lcbt > 0, i.e., if marginal cost of the centralbank providing credit is less than the marginal private cost then it willbe optimal for the central bank to engage in credit policies.


Optimal policy: credit policiesValue of Ξcb,critL for different types of (serially correlated) shocks:Curdia and Woodford Figure 4, p. 69.

Credit policies are optimal when this cutoff exceeds 3.5 under theCurdia-Woodford calibration.



Different shocks all of which increase ω by 1.2%:I Multiplicative shock to Ξp —effects first and second derivative and soworks via direct cost and spread channels.

I Additive χt shocks actually make credit easing by the central bank lessdesirable because this shock reduces L and so ΞpL falls and χ′ falls(acting to reduce ω).

Figure above assumed policy interest rates adjusted optimally —benefits of credit easing increase at the zero lower bound, especiallyin a discretionary policy regime or under a Taylor rule.


Conclusions on Curdia-Woodford

If interest rate policy is optimal, no role for quantitative easing.

Credit policies might be optimal but only if cost of central bankintermediation is not too large.

Source of shocks matter —multiplicative shocks to cost of privatesector intermediation raise marginal cost and spreads and can makecentral bank credit lending optimal.

I More likely to be the case if interest rate policy is discretionary orfollows a Taylor rule and therefore does not display inertia.


Gertler and Kiyotaki (2011)Earlier work focused on frictions affecting non-financial borrowers andtreated intermediaries as a veil. Current crisis centered onintermediaries.Earlier work did not consider the types of unconventional policies usedin the current crisis.Introduce agency problem between borrowers and lenders —creates awedge between external finance cost and opportunity cost of internalfinance.Assume financial intermediaries have skills in evaluating borrowers —makes it effi cient for credit to flow through intermediaries.Households deposit funds with intermediaries, intermediaries lend tonon-financial firms.Agency problem limits ability of intermediary to obtain funds fromdepositors. This accounts for wedge between deposit and loan rates.Spread widens in a crisis which raises cost of funds to non-financialfirms.


Gertler and Kiyotaki (2011)

Current crisis suggests intermediaries also have problems raising fundsfrom other intermediaries, for example in an interbank market. Sothey assume intermediaries are subject to idiosyncratic liquidityshocks. Disruptions in interbank market can affect real activity.Financial markets become segmented and generate an ineffi cientallocation of funds among intermediaries.

Continuum of firms of mass unity located on a continuum of islands.

Investment opportunities arrive randomly to fraction πi of islands. Noopportunity on fraction πn = 1− πi of the islands.

I Only firms with investment opportunities can obtain new capital.

Households deposit funds into banks; bank deposits are riskless oneperiod securities.

Households can also hold riskless one period government debt (aperfect substitute for bank deposits).


The model: banks

Each period, banks choose an island to operate on prior to theresolution of any uncertainty.

During period, bank can only make loans to firms on its island —localized lending.

Because banks can freely move to another island at end of period, exante returns will be equal across islands.

Banks raise funds in a national financial market.I Consists of a retail market in which banks raise funds from householdsand a wholesale market in which they raise funds from other banks.

After retail market closes, investment opportunities are determined(i.e., it is like a limited participation model).


The agency problemAfter receiving deposit funds, banker can transfer fraction θ ofdivertable assets to own use.

I Divertable assets are

Qht sht −ωbht , 0 < ω < 1

I If bank diverts funds, it fails and creditors obtain 1− θ of the funds.I Because of bank’s incentive to divert funds, creditors will limit theamount they lend. This means banks will face a borrowing constraint.

I With ω = 1, interbank market operates frictionlessly in that bankcannot divert any of bht . In this case, banks will not be constrained inborrowing in the interbank market.

Let V (sht , bht , dt ) be value function at the end of period t. Then to

ensure bank does not divert funds, the incentive constraint

V (sht , bht , dt ) = vsts

ht − vbtbht − vtdt ≥ θ

(Qht s

ht −ωbht

)must hold.


Bank’s decision problemLet λht be Lagrangian multiplier on the incentive constraint. Letλt ≡ ∑h=i ,n πhλht be average across states and islands.The FOCs then can be written as

(vbt − vt )(1+ λt ) = θωλt

Marginal cost of interbank borrowing exceeds marginal cost ofdeposits if and only λt > 0 and ω > 0.(

vstQht− vbt

)(1+ λht ) = λht θ(1−ω)

The marginal value of assets in terms of goods, vst/Qht exceeds themarginal interbank borrowing cost if λht > 0 and ω < 1.

vtnht ≥[

θ −(vstQht− vt

)]Qht s

ht − [θω− (vbt − vt )] bht

The value of net worth must be at least as large as weighted measureof the bank’s asset holdings with weights related to the value ofdiverting the asset.


Credit policies: lending facilities

Central banks, unlike financial intermediaries, not constrained inraising funds.

Central bank can intermediate credit by direct lending.

In constrained financial markets, this expands credit supply and totallending.

In unconstrained markets, it simply displaces private lending.


Credit policies: discount window lending

When ω = 0 (symmetric friction), central banks can borrow frombanks with funds by issuing riskless government debt and lend toconstrained banks at rate Rm,t+1.

But this just displaces private interbank lending unless central bankhas advantage in supply funds.

Gertler and Kiyotaki assume central bank is able to enforce repaymentbetter; θ(1−ωg ) < θ.

For both private and central bank interbank lending to occur, centralbank should set Rm,t+1 to make excess cost of discount windowborrowing equal to ωg times excess value of assets on investingislands.

I If banks cannot default to central bank (ωg = 1), then penalty rate ondiscount window borrowing is equal to excess return on assets oninvesting islands.

I Central bank could expand lending to drive excess returns to zero. (GKassume a limit on central bank lending.)


Labor market frictions

1 Sticky wages and labor wedges

1 Implications for policy objectives and effi ciency2 Effects on the long-run Phillips curve

2 Unemployment and the extensive margin

1 Implications for policy objectives2 Effects of labor market structure


Sticky wages

Erceg, Henderson, and Levin (1999) have employed the Calvospecification to incorporate sticky wages and sticky prices into anoptimizing framework.

The goods market side of their model is identical in structure to theone developed earlier.

In the labor market, however, they assume individual householdssupply differentiated labor services; firms combine these labor servicesto produce output.


Sticky wages and pricesImplications

The model of inflation adjustment based on the Calvo specificationimplied that inflation depended on real marginal cost. In terms ofdeviations from the flexible-price equilibrium, real marginal costequaled the gap between the real wage and the marginal product oflabor (mpl). Thus, letting ωt denote the real wage,

πt = βEtπt+1 + κ (ωt −mplt ) . (30)

With flexible wages, workers were always on their labor supply curves;despite price stickiness, nominal wages could adjust to ensure the realwage equaled the marginal rate of substitution between leisure andconsumption (mrs).


Policy objectives with sticky prices and wagesWhen both wages and prices display stickiness, can’t keep x = 0when there are productivity shocks

I If price level is stabilized, sticky wages prevent the real wage fromadjusting, so ineffi cient output variability results.

I Similarly, stabilizing the wage level still leaves prices sticky so realwages cannot jump.

Welfare costs arise from price dispersion and from wage dispersion.Approximation to the welfare of representative agent is now equal tothe expected present discounted value of

Lt =12

(λpπ2t + λwπ2wt + λxx2t

)where πw is wage inflation, λp + λw = 1, and

λwλp

∝κpκw.

Place more weight on stabilizing wage inflation if wages are stickierthan prices.


Unemployment and the business cycle

1965 1970 1975 1980 1985 1990 1995 2000 20056

4

2

0

2

4

6GDP_GAPU_GAP


U. S. unemployment rate and the output gap

6 4 2 0 2 4 63

2

1

0

1

2

3

4

5

6

output gap

cycl

ical

une

mpl

oym

ent

Figure: Civilian unemployment rate minus CBO estimate of NAIRU and cyclicaloutput (HP filtered real GDP), 1964:1-2009:4


Unemployment and optimal monetary policy

Dominant new Keynesian modelI Has provided an important framework for thinking about policy.I Offers structure for conducting welfare-based policy analysis.

But ... the framework has many shortcomings.I Lack of financial frictions or role for financial intermediation.I All labor adjustment at the intensive (hours) margin —under-employment but no unemployment.

Questions:I Does stabilizing unemployment constitute a separate objective aflexible inflation targeter should pursue?

I The canonical monetary policy model cannot cast light on the costs ofineffi cient fluctuations in unemployment relative to the costs ofinflation volatility.


Most cyclical adjustment is at the extensive margin

1965 1970 1975 1980 1985 1990 1995 2000 20056

4

2

0

2

4HOURS_PERSONEMPLOYMENTTOTAL_HOURS

Figure: Employment (blue), hours/worker (black), total hours (green), NFB, logs,HP filtered, 1964:1 -2009:4.


Adding unemployment to DSGE modelsEarly work focused on dynamics:

I Cooley and Quadrini (1999) — limited participationI Walsh (2003, 2005) —price stickiness, examined effect on persistence.

Empirical work:I Ravenna and Walsh (2008) — tested the unemployment-based Phillipscurve;

I Trigari (2004) —estimated DSGE;I Sala, Söderström, Trigari (2008) —estimated US with wage and pricerigidities;

I Christoffel, Kuester, and Linzert (2006) —EU area;I Gertler, Sala, and Trigari (2006) —estimated model with real wagerigidity.

I Christiano, Trebrandt, and Walentin (2010), Galí, Smets and Wouter(2011).

Policy analysis:I Thomas (2008); Blanchard and Galí (2007); Ravenna and Walsh(2011), Christiano, Trabandt, and Walenti (2010).


The DMP model (Hall 2012)

The basic DMP model has three components:

1 Beveridge curve:

∆ut = ρ(1− ut ) + µθat ut , θt =vtut, 0 < a < 1

∆ut = 0⇒ ut =ρ

ρ+ µθat


The Beveridge curve

3 4 5 6 7 8 9 10 111.5

2

2.5

3

3.5

4The U.S. Beveridge Curve

Civilian unemployment rate (%)

Vac

ancy

rat

e (%

)

* 2000:122007:12+ 2008:12009:12x 2010:12012:2

The Beveridge Curveshifted out in theGreat Recession


The DMP model (Hall 2012)

The basic DMP model has three components (steady state version):

1 Beveridge curve;2 A theory of job creation:

V = −κ + q(θ) (µz − w) + [1− q(θ)]V ⇒ (µz − w) = κ

q(θ)

q(θ) (µz − w) = κ


The DMP model (Hall 2012a)

The basic DMP model has three components:

1 Beveridge curve;2 A theory of job creation:3 A bargaining model of wages (typically Nash):

w − R = β (µz − R)⇒ w = (1− β)R + βµz

Problem —the Shimer puzzle


Ravenna and Walsh (2011)

Basic new Keynesian model for households and retail firms

Add a Mortensen-Pissarides search and matching model of the labormarket.

Ignore adjustment on the intensive margin to focus on the extensivemargin.

Wages are flexible and set by Nash bargaining (but bargaining share isstochastic).

I Keeps model very similar to basic NK model, but most recent work alsoassumes real wage stickiness.


Final goods

Household obtain utility from consumption:

U(Ct ) =C 1−σt

1− σ

Consumption consists of market goods and home productions:

Ct = Cmt + wu(1−Nt ).

The total expenditure on final goods from households and wholesalefirms is∫ 1

0Pt (j)Cmt (j)dj + κ

∫ 1

0Pt (j)vt (j)dj = Pt (Cmt + κvt )

Goods market clearing:

Yt = Cmt + κvt


Wholesale goods, employment and wages

Production by wholesale firm i is

Y wit = ZtNit ,

where Zt is a common, aggregate productivity disturbance with amean equal to 1 and bounded below by zero.

Wholesale firms sell their output in a competitive market at the pricePwt .

The real marginal cost of a retail firm is the inverse of the retail-pricemark up:

1µt≡ PwtPt.


The labor market

Wholesale firms must post vacancies to obtain new employees.

If a job produces output Zt and wt is the wage paid to the worker,than the value of a filled job in terms of final goods is

Jt =(PwtPt

)Zt − wt + (1− ρ)βEt

(λt+1λt

)Jt+1,

The job posting condition is qtJt = κ, where κ is the vacancy postingcost and qt is the probability of filling a vacancy, so

Ztµt= wt +

κ

qt− (1− ρ)βEt

(λt+1λt

)(κ

qt+1

)If κ = 0, this yields the standard result that 1/µt = P

wt /Pt = wt/Zt .


Employment dynamics

Each period, an exogenous fraction ρ of existing matches terminate.The number of unemployed job seekers in period t is

ut ≡ 1− (1− ρ)Nt−1.

Unemployed workers are matched stochastically with job vacancies,with matching process is represented by a CRS matching function:

m(ut , vt ) = χv αt u

1−αt = χθα

t ut

where θt ≡ vt/ut is the measure of labor market tightness, and0 < α < 1.

Aggregate employment evolves according to

Nt = (1− ρ)Nt−1 +m(ut , vt ).


Wages and the relative price

The equilibrium real wage under Nash bargaining is

wt = (1− bt )wu + bt[Ztµt+ (1− ρ)

(1Rt

)κEtθt+1

]The relative price of wholesale goods in terms of retail goods is equalto Pwt /Pt = 1/µt = τt/Zt .where

ξt ≡ wu +(

11− bt

)(κ

qt

)−(1− ρ)

(1Rt

)Et

(1− bt+1pt+11− bt+1

)(κ

qt+1

).

Labor market tightness affects inflation through ξt .


Linearized modelThe unemployment-based Phillips curve

The linearized Phillips curve takes the standard form:

πt = βEtπt+1 − δµt .

To obtain a Phillips curve in terms of unemployment gaps, we use thefact that real marginal cost can be expressed as a function of labormarket tightness and ut+1 = ρu ut − αρηθt to obtain

πt = βEtπt+1 − δγ1ut+1 + δγ2 rt + δBbt ,

where the ai are functions of the model’s structural parameters.

There is also a cost channel and bargaining shocks act like costshocks in a basic NK model.


Welfare

The second order approximation to welfare is

∞

∑i=0

βiU(Ct+i ) =U(C )1− β

− ε

2δUc C

∞

∑i=0

βiLt+i + t.i .p.

where t.i .p. denotes terms independent of policy, and the period-lossfunction is

Lt = π2t + λ0 c2t + λ1 θ2t .

The weight on c2t is the same as that obtained in a standard NKmodel if utility is linear in hours worked.

Weight on labor market tightness is

λ1 = (1− α) (δ/ε) (κV/C ).


Distortions: intuition

U(Ct ) = U∫ [

(ct (j))ε−1

ε dj] ε

ε−1+ C non−markett

1 Ineffi cient volatility in consumption when the consumption gap isnon-zero.

2 Ineffi cient composition of market consumption resulting from relativeprice dispersion due to non-zero inflation.

3 Ineffi cient composition of total consumption due to search frictionswhen the labor market tightness gap is non-zero.


Calibration

Table 1: Parameter ValuesExogenous separation rate ρ 0.1Vacancy elasticity of matches α 0.5Replacement ratio φ 0.54Steady state vacancy filling rate q 0.9Labor force N 0.9416Discount factor β 0.99Relative risk aversion σ 2Markup µ 1.2Price adjustment probability 1−ω 0.25


Optimal commitment —response to a bargaining shock

2 4 6 8 10 12 14 16 18 20 22 240.1

00.10.2

inflation

ρb = 0ρb = 0.8

2 4 6 8 10 12 14 16 18 20 22 240

0.10.20.3

unemployment

ρb = 0ρb =0.8

2 4 6 8 10 12 14 16 18 20 22642

0labor market tightness

ρb = 0ρb = 0.8

Figure: Response to a one standard deviation bargaining shock under optimalcommitment (π and θ scaled in percentage point deviations from steady state;unemployment scaled as percentage points of total labor force).


Optimal commitment —role of the loss function

We consider three alternative objectives for the central bank:

1 The welfare based objective:

Lt = π2t + λ0 c2t ++λ1 θ2t .

2 A standard inflation-consumption gap loss function:

Lt = π2t + λ0 c2t

3 An inflation and unemployment gap loss function:

Lt = π2t + λu2t .


Table 2: Alternative policy objectives (optimalcommitment)

Quadratic loss Welfare-based lossRelative to Opt.Commitment (%) Welfare cost∗ σπ σu σθ σπ/σu

0 0 0.24 0.72 11.82 0.33Std. Loss in π and c−gap, λ = λ0

Welfare cost∗ σπ σu σθ σπ/σu4.59 0.0011 0.02 0.75 12.36 0.03

Std. Loss in π and u−gap, λ = 0.0035Welfare cost∗ σπ σu σθ σπ/σu

0.34 0.0001 0.22 0.72 11.86 0.32Std. Loss in π and u−gap, λ = 0.0521

Welfare cost∗ σπ σu σθ σπ/σu275.93 0.0683 1.96 0.51 8.27 3.83

∗ Relative to welfare-based optimal commitment, as percent of steady-stateconsumption.


Optimal commitment —role of the loss function

5 10 15 200

0.51

1.5inflation

5 10 15 20

0.10.20.3

unemployment

5 10 15 20642

labor market tightness, θ

Figure: Responses to one std dev. bargaining shock under optimal commitmentpolicies minimizing different loss functions: o welfare loss; ∗ eq. 32, + eq. 33with λ = 0.0035; x eq. 33 with λ = 0.0521.Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 175 / 319

U.S. versus EU. calibration

Stylized comparison of U.S. and E. U. labor markets.

Hold all parameters the same except rate of job destruction ρ,steady-state employment and replacement ratio.

U.S. values ρ = 0.10 N = 0.9416 φ = 0.54EU values ρ = 0.025 N = 0.0.8989 φ = 0.65


Implied Phillips curves: U.S. and EU

U.S. calibration: πt = βEtπt+1 − 0.087ut+1 + 0.103rt + 0.081btEU. calibration: πt = βEtπt+1 − 0.065ut+1 + 0.845rt + 0.099bt

Two differences are apparent.I Interest rate channel is much larger with EU calibration.I Inflation is less sensitive to the unemployment gap with EU calibration.

These differences reflect the higher persistence of unemploymentunder the EU calibration (ρu = 0.798 for EU versus 0.345 for U.S.)

I When ρu is large, both current and future labor market conditionsmove together; impact of current unemployment offset to some degreeby the co-movement of expected future unemployment.

EU faces better trade off.


Summary

Developed micro-founded Phillips curve and simple policy modelexpressed in terms of inflation and an unemployment gap;

Developed model consistent policy objectives that highlight role oflabor market frictions;

Distortions due to labor frictions can be summarized in terms ofstabilizing a labor market tightness gap;

I This distortion can also be expressed in terms of an unemployment gap.

Ineffi cient fluctuations in unemployment pose policy trade-offsI But price stability remains close to optimal, particularly with EUcalibration.

Focusing on the wrong welfare function can lead to significant welfarelosses.


Search frictions, wedges, and price stability

Why do many models with frictions imply price stability is close tooptimal?

Tax intrepretation. Suppose economy characterized by sticky prices,search frictions in the labor market, and monopolistic competition.

Ravenna and Walsh (2011) show that the first best allocation can besupported using three taxes and monetary policy.

I Price stability that ensures the retail price markup µt is constant.I A steady-state subsidy τµ to retail firms to eliminate distortion due toimperfect competition to ensure the markup is constant at 1.

I A tax (subsidy) τft on intermediate firms to ensure vacancy posting iseffi cient.

I A tax on household labor consumption τCt to ensure hours choice isoptimal.


Tax instruments

If the Hosios condition is satisfied, then the first best can be achievedwith a steady-state subsidy to retail firms to eliminate distortion dueto imperfect competition plus a policy of price stability. The othertwo taxes are not needed.

If the Hosios condition is not met, thenI τft can be used to ensure the markup varies to ensure effi cient vacancyposting.

I This distorts the choice of hours so τCt is needed to ensure hours areeffi cient.

I Monetary policy ensures price stability to eliminate the distoritoncreated by price dispersion.

I τµ ensures µ = 1.


Using only monetary policy

If wage bargaining is Nash but fails the Hosios condition, theintermediate sector tax that corrects firms’incentive to post vacancyis large but basically acyclical.

Monetary policy can replicate this but doing so requires littlemovements in markups, so price stability is close to optimal.

With a wage norm at an ineffi cient level, the required intermediatesector tax must vary significantly to achieve effi cient vacancy posting.

Replicating this using only monetary policy would require largedeviations from price stability and distort the hours choice, so thecost of trying to eliminate ineffi cient vacancy postings is large.Optimal policy improves only a little relative to price stability.


Is the long-run Phillips curve vertical?

Akerlof, Dickens, and Perry (1996)

Bengino and Ricca (2011)

Coibion, Gorodichenko, and Wieland (2011).


Resource mobility

Understanding costly resource mobility is important:I In U.S. — for understanding whether recent high unemployment isstructural in nature because of the inability of labor resources to shiftquickly between uses.

I In EU — for understanding how the flow of resources among membercountries affects EU-wide developments and inflation

DSGE policy models:I Costly to adjust prices but labor and capital can move between firmswithout cost.

I Prominent examples: Smets and Wouter (2003, 2007), Christiano,Eichenbaum and Evans (2005).

I Models of currency union: Benigno (2004), Galí and Monacelli (2008)

F Perfectly integrated financial markets, perfect mobility of labor withinmembers but absolute immobility across member states.


Key questions

1 How important is resource mobility for the transmission mechanism ofmonetary policy?

2 How important is resource mobility for the objectives of monetarypolicy?

Resource mobility will matter for both what monetary policy can doand should do.

Focus will be on labor mobility to illustrate this conclusion.


Sectoral dispersion and unemployment

Sectoral dispersion and unemployment was a topic of debate in the1980s.

I Lilien (1982)I Abraham and Katz (1986)

Lilien’s index of dispersion:

σt =

[K

∑i=1

(ei ,tet

)(∆ log ei ,t − ∆ log et )

2

]1/2

.

Shifting Beveridge Curve has reopened debate.


Sectoral dispersion, unemployment, and vacancies (U.S.)

1985 1988 1991 1994 1997 2000 2003 2006 20090

2

4

6

8

10

12

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5U N R ATEVAC R ATELIL IEN _SIGMA

Figure: The civilian unemployment rate, the vacancy rate, and sectorial dispersion(right scale); monthly, U.S. data, 1985:1-2010:1. The dispersion measure is a12-month moving average.Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 186 / 319

Sectoral dispersion, unemployment, and vacanciesAbraham and Katz (1986) regressions

Table 1AU.S.: Monthly 2000:12-2010:09

zt= c + a0σt+a1σt−1+b1zt−1+∑4i=1 ci ipt−i

Unemployment rate Vacancy rate

a0 0.31∗∗ −0.16a1 −0.29∗∗ 0.09b1 1.01∗∗ 0.82∗∗

∑4i=1 ci 0.01 −0.00

∗∗ Significant at the 5% level; ∗ Significant at the 10% level.


Sectoral dispersion, unemployment and vacanciesAbraham and Katz (1986) regressions

Table 1BU.S.: Monthly 2000:12-2010:09

∆zt= c + a0∆σt+∑4i=1 ci∆ipt−i

Unemployment rate Vacancy rate

a0 0.24∗∗ −0.04a1b1∑4i=1 ci −0.17∗∗ 0.06∗

∗∗ Significant at the 5% level; ∗ Significant at the 10% level.


Sectoral dispersion

Using JOLTS data, sectoral dispersion is associated with higherunemployment, consistent with Lilien’s earlier findings.

Vacancies are negatively (but not statistically significantly) related tosectoral dispersion;

I This is evidence that the sectoral dispersion index is just reflectingcyclical factors;

I But, weaker evidence against Lilien’s hypothesis than found byAbraham and Katz.

Even if sectoral shifts do not raise the natural rate of unemployment,labor reallocations across firms, sectors, and time can be costly —withimplications for macro dynamics and policy.


Models of costly labor allocation: three examples

1 Quadratic costs of adjusting employment;2 Search with skill heterogeneity —composition effects;3 Search model with two sectors —costly sectoral reallocation.


Example 1: Quadratic costs of adjusting laborLechthaler and Snower (2011)

If it is costly for firms to adjust their employment levels, then hiringdecisions and labor utilization decisions will need to be forwardlooking, just as price setting behavior is.

This also means that these adjustment costs can affect marginal costsand inflation.

This affects the way the economy responds to disturbances —i.e.economic dynamics are affected.

Volatile employment generates costs that monetary policy can affect.

Implies stabilizing employment changes is a legitimate objective ofmonetary policy along side inflation and output gap stabilization.


Response to a markup shock: optimal commitment

0 2 4 6 8 10 12 14 161.5

1

0.5

0Output gap (commitment)

0 2 4 6 8 10 12 14 160.2

00.20.40.60.8

Inflation (commitment)

Ψ=0Ψ=1.85Ψ=4

Figure: Optimal response under commitment to a serial correlated markup shockin the quadratic costs of adjustment model.


Example 2: Skill heterogeneity?Decline in vacancy yield

2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

2

4

6

8

10

12

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9UVYIELD

Figure: The U.S. unemployment rate, the vacancy rate, and the hiring yield (rightscale).


Decline in vacancy yield

2001 2002 2003 2004 2005 2006 2007 2008 2009 20101.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Figure: The hiring yield and forecasted yield based on labor market tightness(V/U). Forecast obtain from an OLS regreesion of the yield on a constant andV/U, 2000:12 - 2009:12.


Example 2: Skill heterogeneityRavenna and Walsh (2010)

Low skill and high skill workers.

Low skill worker more likely to experience job separation.

In a recessions, the skill mix of the unemployed shifts towards low skillworkers:

I Reduces the vacancy yield rate as firms see more job applicants theydon’t want to hire;

I Reduces incentive for firms to post vacancies;I Job finding rate falls because probability of finding a job for a low-skillworker falls and because low-skill workers become a larger share of thetotal unemployed.


Response to a neutral TFP shock

1 2 3 4 50

0.2

0.4

0.6

0.8Percentage points change unemployment rate

1 2 3 4 50

2

4

6Percentage points change unemployment rate low skill

Neg

ativ

e pr

oduc

tivity

sho

ck

1 2 3 4 50

0.1

0.2

0.3

0.4Percentage points change unemployment rate high skill

DOTTED: BASELINE SOLID: LARGE LABOR FLOWS


Role of composition effect

1 2 3 4 55

4

3

2

1

0Ne

gativ

e pr

oduc

tivity

sho

ck

Job finding probability logchange EU

unconditionalhighskilllow skill

1 2 3 4 50

20

40

60

80Unconditional screening out rate logchange EU

baselinewithout composition effect

Figure: Skill heterogeneity: response to a negative productivity shock: Job findingand screening ratesCarl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 197 / 319

Role of skill-bias shock

1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4Percentage points change unemployment rate

1 2 3 4 50

1

2

3

4

5

6

7Percentage points change unemployment rate low skill

1 2 3 4 50

0.05

0.1

0.15

0.2

0.25Percentage points change unemployment rate high skill

Ski

llbi

ased

neg

ativ

e TF

P s

hock

DOTTE D: Skillbiased shock1 2 3 4 5

1.4

1.2

1

0.8

0.6

0.4

0.2

0Retail output

SOLID: Aggregate shock


Example 3: Sector heterogeneity

Two sectors, hiring costs are higher if worker previously employed inthe other sector:

I captures the idea that workers may have sector or job specific skills;I implies the composition of the pool of unemployed matters for jobcreation.

Burst of unemployment in one sector pushes up unemployment butmay weaken incentives for firms in other sectors to create jobopenings.

Effi ciency implications —employment reduction in one sector imposesa negative externality on firms in other sectors as average quality ofthe unemployed (from the perspective of other sectors) deteriorates.


Sector heterogeneity and costly labor searchA common productivity shock

5 10 15 200

0.05

0.1

0.15

0.2y

5 10 15 200.8

0.6

0.4

0.2

0h

5 10 15 200.4

0.3

0.2

0.1

0n

5 10 15 206

4

2

0v

5 10 15 200

0.5

1

1.5

2u

5 10 15 200.2

0.15

0.1

0.05

0infl

Figure: Impulse responses to a serially correlated productivity shock to bothsectors: Solid line is model without composition effects.


Sector heterogeneity and costly labor searchA sector specific productivity shock

5 10 15 200.2

0

0.2

0.4

0.6

0.8h1

5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12h2

5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35n1

5 10 15 200.1

0.05

0

0.05

0.1n2

Figure: Impulse responses of hours and employment to a negative productivityshock only to sector 1: Solid line is model without composition effects.


Is the long-run Phillips curve vertical?

Akerlof, Dickens, and Perry (1996)

Bengino and Ricca (2011)

Coibion, Gorodichenko, and Wieland (2011).


Summary and implications

Current DSGE policy models minimize costs of labor reallocation.

The Great Recession in the U.S. does not overturn earlier conclusionsabout the link between sectoral dispersion and unemployment.

I Evidence from Beveridge Curve and decline in vacancy yield suggestsmismatch of workers and job openings may have increased.

When labor reallocation is costly, the economy’s dynamics and thecost of fluctuations are affected.

I Role for labor market objectives.I Low turnover in labor markets can raise the importance of inflationstability.

I Composition effects may be important for macro dynamics andtherefore for policy objectives and for designing monetary policy.

These general conclusions will apply to other factors of productionand to other situations in which there are costs of adjustment thatreflect the imperfect mobility of resources.


New Keynesian small open economy: Adolfson, Laséen,Linde, and Villani (2008)

Four different types of firms: firms producing (1) domestic good, (2)importing consumption goods, (3) importing investment goods, and(4) exporting goods.

Within each category, continuum of firms each producing adifferentiated product.

Domestic goods firms use labor and capital to produce output whichthey sell to a retailer.

Retailer transforms domestic goods into a homogeneous final goodsold to households.

Relative prices: (1) domestic goods and final good; (2) domesticgoods and imported consumption goods; (3) consumption andinvestment goods; (4) final good and export goods.


Adolfson, Laséen, Linde, and Villani (2008)New Keynesian small open economy

The Ct that generates utility is a basket of domestically producedgoods and imported consumption goods:

Ct =

[(1−ωc )

1ηc

(C dt) ηc−1

ηc +ω1

ηcc (Cmt )

ηc−1ηc

] ηcηc−1

So domestic and imported goods are imperfect substitutes.

Households own the capital stock:

Kt+1 = (1− δ)Kt + Υt

[1− S

(ItIt−1

)]It

They also supply labor and set wages ala Calvo, indexing to theinflation target and productivity.


Adolfson, Laséen, Linde, and Villani (2008)Retails bundle domestic goods via standard Dixit-Stiglitz CESaggregator.Leads to demand functions facing domestic firm i and price aggregateof form

Yi ,t =

(Pdi ,tPt

) λdtλdt −1

Yt ; Pt =[∫ 1

0

(Pdi ,t) 11−λdt di

]1−λdt

λdt is time varying —markup is 1/λdt so markup can vary.Production function of domestic firms:

Yi ,t = z1−αt εtK α

i ,tH1−αi ,t − ztφ

where z is a unit root technology shock and ε is a stationaryaggregate productivity shock. K is capital; H labor.ALLV assume a fraction of a firm’s wage bill must be financedthrough loans — ignore this.


Adolfson, Laséen, Linde, and Villani (2008)New Keynesian small open economy

Continuum of importing consumption and investment firms; eachbuys a homogeneous good at price P∗t in world markets andtransforms it into a differentiated good sold in the domestic economy.

I Marginal cost for these firms is StP∗t , where St is the nominalexchange rate (domestic currency per unit of foreign currency)

Exporting firms buy the homogeneous domestic final good andconvert it into a differentiated export good.

I Marginal cost for these firms is Pdt /St ; to generate one unit ofdomestic currency, the exporter needs to sell goods worth 1/St inforeign currency which cost Pdt /St to purchase.

All firms have sticky prices and time varying markups.I Leads to four Phillips curves (for domestic goods, importedconsumption goods, imported investment goods, and exported goods.

I Allows for imperfect pass through.


Adolfson, Laséen, Linde, and Villani (2008)

Domestic firms subject to Calvo price adjustment mechanism: firmsadjust optimally with probability 1− ξd . With probability ξd , firmspartially index to a combination of the central bank’s inflation targetπct and last period’s inflation rate: Pi ,t =

[πκdt−1 (π

ct )1−κd

]Pi ,t−1.

In terms of log-deviations from steady state,

πdt − πct =

(β

1+ κd β

)Et(

πdt+1 −πct+1)

+

(κd

1+ κd β

)(πdt−1 −πct−1

)+

(κd β

1+ κd β

)(Etπct+1 − πct )

+(1− ξd ) (1− βξd )

ξd (1+ κd β)

(mct + λ

dt

)


Clarida, Galí, and Gertler (2001), Galí and Monacelli(2006)

Simplified small open economy: drop investment.

The model consists of households who supply labor, purchase goodsfor consumption, and hold money and bonds, and firms who hirelabor and produce and sell differentiated products in monopolisticallycompetitive goods markets.

I consumption risk is shared internationally.

Each firm set the price of the good it produces, but not all firms resettheir price each period.

Households and firms behave optimally: households maximize theexpected present value of utility and firms maximize profits.


Household utility

Households consume a CES composite of home and foreign goods,defined as

Ct =[(1− γ)

1a

(C ht) a−1

a+ γ

1a

(C ft) a−1

a

] aa−1

for a > 1.

C h (and C f ) are Dixit-Stiglitz aggregates of differentiated goodsproduced by domestic (and foreign) firms.

Demand for good produced by firm j :

cjt =(PjtPt

)−θ

C ht ; Pt =[∫ 1

0p1−θjt dj

] 11−θ


Household utility

Let Pht (Pft ) be the average price of domestically (foreign) produced

consumption goods. The problem of minimizing the costPht C

ht + P

ft C

ft of achieving a given level of Ct yields:

C htC ft

=

(1− γ

γ

)(PhtP ft

)−a.

The aggregate price index is

Pct ≡[(1− γ)

(Pht)1−a

+ γ(P ft)1−a] 1

1−a.

is the aggregate (consumer) price index.


Household utility

Household utility depends on its consumption of the composite good,money holdings, and on its labor supply:

Et∞

∑i=0

βi

[C 1−σt+i

1− σ+

γ

1− b

(Mt+i

Pt+i

)1−b− χ

N1+ηt+i

1+ η

].

Intertemporal optimization implies the standard Euler condition,

C−σt = βEtRt

(PctPct+1

)C−σt+1,

Optimal labor-leisure choice requires that the marginal rate ofsubstitution between leisure and consumption equal the real wage.This condition takes the form

Nηt

C−σt

=Wt

Pct.


Law of one price

Assume that the law of one price holds. This implies that

P ft = StP∗t ,

where P∗t is the foreign currency price of foreign-produced goods andSt is the nominal exchange rate (price of foreign currency in terms ofdomestic currency).

This specification assumes complete exchange rate pass-through;given P∗t , a 1% change in the exchange rate produces a 1% change inthe domestic currency price of foreign produced goods P ft .


Law of one price and terms of tradeGiven that we have assumed the law of one price holds, the pricelevels in the domestic and foreign countries are linked by

Pct = StPc∗t .

For the rest of the world, we ignore the distinction between the CPIand the price of domestic production, so Pc∗t = P∗t .The terms of trade is equal to the relative price of foreign anddomestic goods:

∆t ≡P ftPht

=StP∗tPht

. (31)

The real exchange rate is defined as the relative price of foreignproduced goods (in terms of domestic currency) relative to the homecountry’s consumer price index:

Qt =StP∗tPct

. (32)


International risk sharing

Assume agents in both economies having access to a complete set ofinternationally traded securities.

The time t home currency price of a bond that pays off one unit ofthe domestic currency at time t + 1 is R−1t , and the Euler conditioncan be written as

βEt

(PctPct+1

)(Ct+1Ct

)−σ

=1Rt.

Since residents in the rest of the world also have assess to these samefinancial securities, intertemporal optimization implies

C ∗−σt

StP∗t= βEt

(Rt

P∗t+1St+1

)C ∗−σt+1 .


International risk sharing

International risk sharing implies,

βEt

(PctPct+1

)(Ct+1Ct

)−σ

=1Rt= βEt

(PctPct+1

)(QtQt+1

)(C ∗t+1C ∗t

)−σ

.

In turn, this implies

Ct = φQ1σt C∗t . (33)

where φ is a constant of proportionality.

For convenience, adopt φ = 1 as a normalization.

This is consistent with a symmetric initial condition with zero netforeign asset holdings.


Uncovered interest parity

If R∗t is the foreign interest rate, then

βEt

(P∗tP∗t+1

)(C ∗t+1C ∗t

)−σ

=1R∗t.

Linearizing this yields

σ (Etc∗t+1 − c∗t ) = r ∗t − Etπ∗t+1

and doing the same for the domestic Euler condition implies

σ (Etct+1 − ct ) = rt − Etπct+1.


Uncovered interest parity

Using the definition of the terms of trade, we have that

(rt − Etπct+1)− (r ∗t − Etπ∗t+1) = σ (Etct+1 − ct )− σ (Etc∗t+1 − c∗t )= Etqt+1 − qt .

Subtracting the inflation terms from each side and using thedefinition of the real exchange rate yields the condition for uncoveredinterest parity:

rt = r ∗t + Et (st+1 − st ) .The domestic nominal interest rate is equal to the foreign (rest of theworld) nominal interest rate plus the expected rate of depreciation inthe domestic currency.


The foreign country (ROW)

Assume the foreign country is large relative to the home country.

This is taken to mean that it is unnecessary to distinguish betweenconsumer price inflation and domestic inflation in the foreign country,and that domestic output and consumption are equal.

The foreign country’s demand for the home country’s output dependson the terms of trade. Assume that foreign households have the samepreferences as those of the home country (so the demand elasticity isthe same).

The Euler condition for foreign country households implies

y ∗t = Ety ∗t+1 −(1σ

)(i ft − Etπ∗t+1

),

orρ∗t ≡ i∗t − Etπ∗t+1 = σ (Ety ∗t+1 − y ∗t ) . (34)


Open economy expectational IS curve

IS curve becomes

yt = Etyt+1 −(1

σγ

)(it − Etπht+1 − ρ∗

)where σγ ≡ [1− γ(1− w)] /σ and ρ∗ ≡ ρ− σγγ(1− w)Et∆y ∗t+1(ρ = 1/β− 1).

This is the small open economy equivalent to the closed economyexpectational IS curve.

Two primary differences between open and closed economy versions.I The elasticity of demand with respect to real interest rate no longerequal to 1/σ. It equals 1/σγ = [1− γ(1− w)] /σ which depends onthe “openness”of the economy.

I The real rate ρ∗ depends on developments in the real of the world.


Inflation in domestic goods prices

To determine the rate of inflation of domestically produced goods,

πht = βEtπht+1 + κ(wt − pht − εt

).

The real consumption wage is wt − pct , and this is related to the realproduct wage wt − pht by the terms of trade: pct = pht + γδt . Sincehouseholds equate the real consumption wage to the marginal rate ofsubstitution between leisure and consumption,

ηnt + σct = wt − pct = wt − pht − γδt .

Hence, real marginal cost is wt − pht − εt = ηnt + σct + γδt − εt .


Inflation in domestic goods prices

We can now eliminate consumption and the terms of trade to obtainan expression for real marginal cost solely in terms of domestic outputand foreign variables.

Since yt = nt + εt , yt = ct + (γw/σ) δt , and yt = y ∗t + (1/σγ) δt ,

πht = βEtπht+1 + κ (η + σγ) (yt − yt ) ,

whereyt ≡ [(σ− σγ)y ∗t − (1+ η)εt ] / (η + σγ)


Parallels with the closed economy NK model

Define the output gap as

xt ≡ yt − yt . (35)

Then we can rewrite the Euler condition as

xt = Etxt+1 −(1

σγ

)(it − Etπht+1 − ρt

)(36)

πht = βEtπht+1 + κ(η + σγ)xt . (37)

whereρt = ρ∗ + σγEt (yt+1 − yt ) . (38)


Policy objectivesThe SOE model, as Clarida, R., J. Galí, and M. Gertler (2001) haveemphasized, is isomorphic to the closed economy new Keynesianmodel.The parallel is even stronger if the central bank’s objective can berepresented as minimizing a quadratic from of the output gap and theinflation rate of domestically produced goods πht .In this case, the central bank’s policy involves minimizing

Et∞

∑i=0

βi[(

πht+i

)2+ λx2t+i

]subject to the inflation adj. eq. and Euler eq.All the conclusions about policy reached there would apply withoutmodification to the small open economy.The critical requirement is that the inflation rate appearing in thecentral bank’s objective function must be πh and not the inflationrate in consumer prices πc .


Policy implications

Policy trade-offs only arise from inflation shocks as long as GDPinflation and not CPI inflation enters the objective function of thecentral bank.

I If CPI inflation matters, trade-offs are more complicated. Anappreciation reduces firms’marginal costs and reduces GDP inflation.

I In face of a positive shock to aggregate spending, the central bankmust raise the nominal interest rate to stabilize the output gap. Butthis leads to an appreciation of the exchange rate and a decline in CPIinflation.


Optimal policy

Distortions in the closed economy:I Steady-state markup and sticky pricesI Use fiscal subsidy to deal with first, price stability to deal with second.

Distortions in the open economy:I Steady-state markup, sticky prices, and possibility of affecting terms oftrade to benefit domestic consumers affects incentives of central bank.

F A monetary expansion that lowers domestic interest rates causes adepreciation of the currency and lowers prices of home productionrelative to foreign goods. This competitive devaluation increasesdemand for home production, and increases Ct relative to C ∗t .

F A rise in s increases y relative to y ∗ and c relative to c∗.

I In special case of σ = η = γ = 1, can still use fiscal subsidy to dealwith first, domestic price stability to deal with second.


Special case

In the special case of σ = η = γ = 1, the flex-price equilibrium iseffi cient if

1− τ = (1− α)−1(1− 1

ε

).

If τ is set to satisfy this condition, then the optimal policy ensures theoutput gap is zero (output equals the effi cient flex-price level) andfrom the NKPC, domestic inflation is zero.

In this special case, domestic inflation targeting is optimal.


Domestic inflation targeting

Continuing with the special case, output moves with the flex-priceoutput level.

A rise in world output has two effects on domestic output:I A contractionary effect due to expenditure switching as a rise in worldoutput improves the terms of trade (a real appreciation as the price offoreign goods falls as their supply increases). This reduces demand anddomestic production.

I An expansionary effect due to the direct demand effect through higherexports.

I If ω > 1 (ω < 1), contractionary (expansionary) effects dominate.I When ω = 1, terms of trade and domestic output are unchanged in theface of a change in world output under an optimal domestic inflationtargeting policy.

With domestic prices constant, the CPI is given by

pft = α(e ft + p

∗t

)= αs ft .


Second order approximation to utility

In special case of σ = η = γ = 1, losses relative to optimal policy fordomestic representative household is

W = −(1− α

2

)( ε

λ

) ∞

∑t=0

βt[

π2h,t +λ(1+ ϕ)

εx2t

]Taking conditional expectations and letting β→ 1,

V = −(1− α

2

)( ε

λ

) [var(πh,t ) +

λ(1+ ϕ)

εvar(xt )

]


Comparing policiesG-M compare optimal policy to a Taylor rule based on domestic priceinflation (DITR), a Taylor rule based on CPI inflation (CITR), and anexchange rate peg. They calibrate domestic and world productivityshocks using Canadian and U.S. data respectively.In face of a domestic productivity shock, DITR leads to a rise in theCPI (there is a real depreciation due to the productivity shock) whileunder CITR, the domestic price level falls (which requires a negativeoutput gap).Under a peg, responses are similar to under CITR but peg makesdomestic and CPI price levels stationary (as they are under optimalpolicy).Optimal policy involves greater terms of trade and nominal exchangerate volatility than DITR or CITR (but domestic inflation and outputgap are keep equal to zero under the optimal policy).Peg yields large welfare losses as it makes TOT too smooth. DITRdoes better than CITR (but that is presumably because under thecase considered, it is optimal to stabilize the domestic price level).


Deviations from law of one price and UIP: ALLV (2008)

The assumptions of the law of one price and UIP inconsistent withdata.

Introducing new distortions:I Imperfect pass through arises because importers prices are sticky(Monacelli 2003).

I For UIP, ALLV note that risk premia are strongly negatively correlatedwith expected exchange rate changes.

F UIP conditionRt = R∗t + Et∆St+1 +Φt

F They assume risk premium is

Φ (at , St , φt ) = −φs (Et∆St+1 − ∆St )− φa at


Role of deviations from UIP: ALLV (2008)


The monetary union model of Benigno (2004)Two regions and a single monetary authority. Two fiscal authorities.Preferences of household j in region i :

U jt = Et

∞

∑s=t

βs−t[U(C js ) + L

(M js

P is, ξ is

)− V

(y js , z

is

)]and i = H is j ∈ [0, n], i = F if j ∈ [n, 1].Each household is a consumer of all goods and a producer of adifferentiated product.Consumption bundel of household j is

C j ≡

(C jH)n (

C jF)1−n

nn(1− n)(1−n), n ∈ [0, 1]

where σ > 1 and

C jH ≡[(1n

) 1σ(∫ n

0c jt (h)

σ−1σ dh

)] σσ−1

C jF ≡[(

11− n

) 1σ(∫ 1

nc jt (f )

σ−1σ df

)] σσ−1

Elasticity across goods within a region is σ, between CH and CF it’s 1.


The model: prices

Price of consumption bundel for region i :

P i =(P iH)n (

P iF)1−n

where

P iH =[(1n

)(∫ n

0pit (h)

1−σdh)] 1

1−σ

P iF =[(

11− n

)(∫ 1

npit (f )

1−σdf)] 1

1−σ

pi (h) is the price of good h sold in region i .

No transportation costs and pH (h) = pF (h) and pH (f ) = pF (f )which implies PH = PF .

Terms of trade T = PF /PH .


The model: demand for individual goods

For goods h and f ,

c j (h) =[p(h)PH

]−σ

T 1−nC j ; c j (f ) =[p(f )PF

]−σ

T−nC j

Fiscal authorities only purchase goods from its own region and hasdemand elasticity σ.

Hence, total demand for goods h and f are

yd (h) =

[p(h)PH

]−σ [T 1−nCW + GH

]; yd (f )

=

[p(f )PF

]−σ [T−nC j + GF

]where CW =

∫ 10 C

jdj .


The model: asset markets

Complete within regions; incomplete across regions.

Within each region, indentical initial wealth, perfect risk sharing —representative household

Internationally traded nominal bond with interest rate Rt . Eulerequation is

UC (Cit ) = (1+ Rt ) βEt

[PtPt+1

UC (Cit+1)

], i = H,F

Aggregate resource constraint for each region:

C it +B it

Pt (1+ Rt )=B it−1Pt

+ CWt .

Assumption that B it−1 = 0 and unitary elasticity of substitutionimplies CW = CH = CF .


The flexible-price equilibrium

The following conditions characterize the equilibrium under flexibleprices in terms of log deviations around the steady state:

CWt =

(η

ρ+ η

)(YWt − gwt

)YWt =

(η

ρ+ η

)YWt +

(ρ

ρ+ η

)gwt

Rt =(

ρη

ρ+ η

)Et[(YWt+1 − YWt

)−(gWt+1 − gWt

)]Tt =

(η

1+ η

)(gRt − Y Rt

)where Y w and gW are world productivity and government purchaseshocks and Y R and gR are relative shocks ( e.g., Y R = Y F − Y H soa relative productivity shock to F reduces PF relative to PH ).


The sticky-price equilibrium

The following conditions characterize the equilibrium under flexibleprices in terms of log deviations around the steady state:

CWt = Et CWt+1 −(1ρ

)(Rt − EtπWt+1

)Y H = (1− n) Tt + CWt + gHt ; Y

F = −nTt + CWt + gFt

In terms of gaps,

cWt = CWt − CWt = yWt = EtcWt+1 −(1ρ

)(Rt − EtπWt+1 − Rt

)


The sticky-price equilibrium

Inflation rates:

πHt = βEtπHt+1 + kHC y

Wt + (1− n)kHT

(Tt − Tt

)= βEtπHt+1 + k

HC y

Ht + (1− n)

(kHT − kHC

) (Tt − Tt

)πFt = βEtπFt+1 + k

FC y

Wt +−nkFT

(Tt − Tt

)= βEtπFt+1 + k

FC y

Ft +−n

(kFT − kFC

) (Tt − Tt

)Tt = Tt−1 + πFt − πHt

Looks like the wage-price block when wages and prices are sticky.


WelfareAssume a fiscal subsidy to offset the distortions due to monopolisticcompetition.Benigno then shows that the second-order approximation to thewelfare of the representative household is

Wt = −ΩE0∞

∑t=0

βtLt

Lt = Λ(yWt)2+ n(1− n)Γ

(Tt − Tt

)2+ γ

(πHt

)2+(1− γ)

(πFt

)2+ t.i .p.+ o

(‖ ξ ‖3

)Union-wide output gap, terms of trade gap, and inflation rates ineach region matter.

I Movements in Tt reduce welfare if they deviate from Tt . Relativeprices in the two regions should move to reflect changes in T .

Inflation enters as weighted average of inflationsquared in two regions.Carl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 240 / 319

Welfare

Basic intuition similar to that with sticky wages and prices, or anytwo sector model with differing degrees of nominal rigidity.

With multiple nominal rigidities, the single instrument of monetarypolicy cannot eliminate all distortions.

Stabilizing a measure of inflation does not ensure that relative pricescan adjust appropriately.

Will be optimal to focus most on


Which inflation rate?

The loss function involves

γ(

πHt

)2+ (1− γ)

(πFt

)2where

γ =ndH

ndH + (1− n)dF

dH =αH

(1− αH )(1− βαH )

dF =αF

(1− αF )(1− βαF )


Which inflation rate?

If prices are equally sticky in each region, αH = αF and γ = n.Weight related to size.

I In this case, if πR ≡ πF − πH ,

γ(

πHt

)2+ (1− γ)

(πFt

)2=(

πW)2+ n(1− n)

(πRt

)2.

If αH = αF , optimal to set πW = nπH + (1− n)πF = 0 sincerelative prices are out of the monetary authority’s control.

πW = πHICP : harmonized index of consumer price inflation.

If prices are flexible in one regions (say F ), then dF = 0 and γ = 1 —welfare only depends on H inflation.


Currency union (Galí and Monacelli JIE 2008)

Currency union consists of a continuum of small open economics.

Each individual country, indexed by i ∈ [0, 1], is of measure zero.Shared preferences, technologies and market structure.

Preferences of household


Households

Representative household of country i maximizes

E0∞

∑t=0

βtU(C it ,Nit ,G

it ); U(C

it ,N

it ,G

it ) = (1−χ) logC it +χ logG it −

(N it)1+ϕ

1+ ϕ

where G it is a measure of public consumption and

C it ≡(C ii ,t)1−α (

C iF ,t)α

(1− α)(1−α)αα, α ∈ [0, 1]

where

C ii ,t ≡(∫ 1

0C ii ,t (j)

ε−1ε dj

) εε−1

and j ∈ [0, 1] is in index of the type of good.α is the weight of imported goods in private consumption. α < 1implies home bias; α can be viewed as an index of openness.


HouseholdsIn addition, C iF ,t is an index of imported goods consumed by countryi , and

C iF ,t ≡ exp∫ 1

0c if ,tdf , c

if ,t ≡ logC if ,t

where C if ,t is the index of country i’s consumption of goods importedfrom country f and

C if ,t ≡(∫ 1

0C if ,t (j)

ε−1ε dj

) εε−1.

Budget constraint is∫ 1

0P it (j)C

ii ,t (j)dj+

∫ 1

0

∫ 1

0P ft (j)C

if ,t (j)djdf +Et

(Qt ,t+1D it+1

)≤ D it +W i

tNit −T it

where P it (j) and Pft (j) are prices (in the single currency). D

it+1 is the

nominal payoff in t + 1 of portfolio held at end of t.Complete set of contingent claims markets and Qt ,t+1 is stochasticdiscount factor for one-period ahead nominal payoffs —commonacross countries.


Household choices

Linearized versions of labor supply and Euler conditions

w it − pic ,t = c it + ϕnit − log (1− χ)

c it = Etc it+1 −(r ∗t − Etπic ,t+1 − ρ

)where ρ ≡ − log β. These hold for all i .

pic ,t is log consumer price index for country i .I Consumer price index is

pic ,t = (1− α)pit + αp∗t

where for each individual country, p∗t is log index of imported goodsprices.


Definitions

Bilateral terms of trade:

S if ,t ≡P ftP it

which is the price of country f ’s domestically produced goods interms of country i’s.

Effective terms of trade for country i :

S it ≡P∗tP it= exp

∫ 1

0

(pft − pit

)df = exp

∫ 1

0s if ,tdf

where s if ,t = log Sif ,t . In logs, s

it =

∫ 10 s

if ,tdf .

ThenP ic ,t = P

it

(S it)α ⇒ pic ,t = p

it + αs it (39)

andπic ,t = πit + α∆s it


International risk sharing and gov’t purchases

Assume symmetric initial conditions with zero asset holdings,

c it = cft + (1− α)s if ,t

and then integrating over f

c it = c∗t + (1− α)s it ; c

∗t ≡

∫ 1

0c ft df (40)

Government buys individual goods in same proportions as households:

G it ≡(∫ 1

0G it (j)

ε−1ε dj

) εε−1

Government only buys domestic goods.


FirmsContinuum of firms in each country with technology

Y it (j) = AitN

it (j)

Real marginal cost is the same for all firms in i

mc it = log(1− τi ) + w it − pit − ait

where τi is an employment subsidy.

Aggregate output and employment (to first order) given by

y it = nit + a

it

Calvo with 1− θ firms resetting prices each period and optimal pricesetting rule is

pit = µ+ (1− βθ)∞

∑k=0

(βθ)kEt(mc it+k + p

it+k

)(41)


Market clearingFor good j produced in country i :

Y it (j) = C ii ,t (j) +∫ 1

0C fi ,t (j)df + G

it (j)

=

(P it (j)P it

)−ε [C it(S it)α+ G it

](42)

Integrating (42) over j , and taking log linear approximation yields

y it = (1− γ)(c it + αs it

)+ γg it (43)

where x is a deviation from steady state and γ ≡ G/Y .But c it = c

∗t + (1− α)s it and s

it = p

∗t − pit , so

y it = (1− γ) c∗t + (1− γ)(pit − p∗t

)+ γg it (44)

Integrating over i yields

y ∗t = (1− γ) c∗t + γgt (45)


InflationNKPC:

πit = βEtπit+1 + λmc it (46)

Marginal cost is

mc it =(w it − pic ,t

)+(pic ,t − pt

)− ait + log(1− τi )

= c it + ϕnit + αs it − ait + log(1− τi )− log(1− χ) (47)

Use (47), y it = (1− γ)(c it + αs it

)+ γg it and y

i = ni + ai to obtain

mc it =(

11− γ

+ ϕ

)y it −

(γ

1− γ

)g it − (1+ ϕ)ait (48)

I Given output, a rise in government spending generates an appreciationwhich lowers the real product wage and so lowers marginal cost.

NKPC for entire union is

π∗t = βEtπ∗t+1 + λmc∗t

= βEtπ∗t+1 + λ

(1

1− γ+ ϕ

)y ∗t − λ

(γ

1− γ

)g ∗t − λ(1+ ϕ)a∗t


With sticky prices

Let y be the deviation of y from the flexible-price, effi cientequilibrium levels (in which y it = a

it and g

it = γ+ ait) where a

it is

country i productivity shock and let ft = gt − yt be the fiscal gap(equal to zero in the effi cient equilibrium).

Then from

mc it = (1+ ϕ) y it −(

γ

1− γ

)f it

Then

πit = βEtπit+1 + λ (1+ ϕ) y it −(

λγ

1− γ

)f it

For the union wide,

π∗t = βEtπ∗t+1 + λ (1+ ϕ) y ∗t −(

λγ

1− γ

)f ∗t


With sticky prices

The fiscal gap acts like a cost shock and forces policy trade offsbetween union wide price stability and maintaining a zero output gap.

If the fiscal gap is zero, no trade off is faced by the union central.

If the individual country decisions about fiscal policy causes f it todeviate from zero, then it is just like a cost shock.


Optimal monetary and fiscal policySecond order approximation to the average utility loss of unionhouseholds due to fluctuations around the effi cient steady state is

W ' 12

∞

∑t=0

βt∫ 1

0

[ε

λ(πit )

2 + (1+ ϕ)(y it )2 +

γ

1− γ(f it )

2]di + t.i .p.

Policy problem:

min12

∞

∑t=0

βt∫ 1

0

[ε

λ(πit )

2 + (1+ ϕ)(y it )2 +

γ

1− γ(f it )

2]di

subject to

πit = βEtπit+1 + λ (1+ ϕ) y it −(

λγ

1− γ

)f it

∆y it −∆y ∗t −(

γ

1− γ

) (∆f it − ∆f ∗t

)+[(

πit − π∗t)+(∆ait − ∆a∗t

)]= 0

and aggregation definitions (i.e., π∗t =∫

πitdi , y∗t =

∫y itdi and

f ∗t =∫f it di .)


First order conditions

Let ψiπ,t be Lagrangian multipliers on the NKPC for country i .

Let ψiy ,t be Lagrangian multipliers on (??) for country i .Let ψ∗π,t , ψ∗y ,t , and ψ∗f ,t be the multipliers on the definitions of theaggregates.



For πit :( ε

λ

)πit + ∆ψiπ,t + ψiy ,t − ψ∗π,t = 0

For y it : (1+ ϕ) y ∗t − λ (1+ ϕ)ψiπ,t + ψiy ,t − βψiy ,t+1 − ψ∗y ,t = 0

For f it : fit + λψiπ,t − ψiy ,t + βψiy ,t+1 −

(1− γ

γ

)ψ∗f ,t = 0

For π∗t : −∫

ψiy ,tdi + ψ∗π,t = 0

For y ∗t : −(1− βL−1

) ∫ψiy ,tdi + ψ∗y ,t = 0

For f ∗t :(

γ

1− γ

) (1− βL−1

) ∫ψiy ,tdi + ψ∗f ,t = 0



FOCs (timeless precommitment) imply( ε

λ

)π∗t +

∫ 1

0∆ψiπ,tdi ⇒= επ∗t + ∆y ∗t = 0

Andf ∗t + λ

∫ψiπ,tdi ⇒ f ∗t − y ∗t = g ∗t = 0.

So επ∗t + ∆y ∗t and f ∗t = y ∗t and aggregate NKPC imply that ify ∗0 = 0,

π∗t = f∗t = y

∗t = 0.

So union wide policy is zero inflation, zero output gap, and zero fiscalgap.


For union member

The second, third, fifth, and sixth FOC imply

(1+ ϕ)y it + fit = λϕψiπ,t

So, since ψiπ,t > 0 if prices are sticky, setting yit = f

it = 0 for each

country is not feasible as an equilibrium under sticky prices.


Implications

Optimal monetary and fiscal policy from the union perspective wouldalways maintain zero inflation, output gap, and fiscal gap. However,this does not mean inflation, output gap, and the fiscal gap are zerofor all i .

Consider an asymmetric productivity shock:I With flexible prices, union wide output gap and fiscal gaps could bekeep at zero.

I Individual countries maintain output and government spending atfirst-best levels while terms of trade adjust in response to relativeproductivity.

I With sticky prices, relative productivity shocks can’t generate requiredmovement in terms of trade (because prices are sticky and theexchange rate is effectively fixed for members of the currency union).

I To increase demand in countries with high productivity shocks (i.e., toboost demand to match supply), fiscal policy must expand.

I Three distortions (output gap, fiscal gap, and inflation), but individualcountries only have one instrument —fiscal policy.


Fiscal and monetary interactions

1 Fiscal or monetary dominance2 Fiscal policy in the new Keynesian model3 Optimal monetary and fiscal policies.


Fiscal and monetary interactions

Several alternative assumptions possible about the relationshipbetween monetary and fiscal policies.

I Fiscal policy assumed to adjust to ensure the government’sintertemporal budget is always in balance, while monetary policy is freeto set the nominal money stock or the nominal rate of interest —described as a Ricardian regime (Sargent 1982), monetary dominance,or one with fiscal policy passive and monetary policy active (Leeper1991).

I The fiscal authority sets its expenditure and taxes without regard tointertemporal budget balance. Seigniorage must adjust to ensureintertemporal budget constraint is satisfied. Case of fiscal dominance(or active fiscal policy) and passive monetary policy.


Fiscal and monetary interactionsThe fiscal theory of the price level

The government’s intertemporal budget constraint may not besatisfied for arbitrary price levels. Following Woodford (1995), theseregimes are described as non-Ricardian. The intertemporal budgetconstraint is satisfied only at the equilibrium price level, and thegovernment’s nominal debt plays a critical role in determining theprice level.


Intertemporal budget balance and seigniorage

The intertemporal budget constraint implies that any governmentwith a current outstanding debt must run, in present value terms,future surpluses.

One way to generate a surplus is to increase revenues fromseigniorage.

Let s ft ≡ tt − gt be the primary fiscal surplus excluding seignioragerevenue.

The government’s budget constraint can be written as

bt−1 = R−1∞

∑i=0R−i s ft+i + R

−1∞

∑i=0R−i st+i . (49)

The current real liabilities of the government must be financed by, inpresent value terms, either a fiscal primary surplus R−1 ∑∞

i=0 R−i s ft+i

or by seigniorage.


Intertemporal budget balance and seigniorageUnpleasant arithmetic

Sargent and Wallace (1981) —“unpleasant monetarist arithmetic” ina regime of fiscal dominance:

I If the present value of the fiscal primary surplus is reduced, the presentvalue of seigniorage must rise to maintain intertemporal budgetbalance.

I Reducing inflation now can mean higher inflation in the future.


Intertemporal budget balance and the nominal interest rate

Suppose that the initial nominal stock of money is set exogenously bythe monetary authority. Does this mean that the price level isdetermined solely by monetary policy, with no effect of fiscal policy?

Fiscal policy can still affect the initial equilibrium price level, evenwhen the initial nominal quantity of money is given and thegovernment’s intertemporal budget constraint must be satisfied at allprice levels.

It does so if it affects the equilibrium real rate of interest.


Intertemporal budget balance and the nominal interest rateExample

Assume perfect foresight equilibrium.

The government’s budget constraint must be satisfied and the realdemand for money must equal the real supply of money.

Assume money demand is

Mt

Pt= f (Rm,t ), (50)

Given Rm , (50) implies a proportional relationship between the M andP. If the initial money stock is M0, then the initial price level isP0 = M0/f (Rm).



The government’s budget constraint given by

gt + rbt−1 = tt + (bt − bt−1) +mt −(

11+ πt

)mt−1. (51)

Consider a stationary equilibrium. The budget constraint becomes

g +(1β− 1)b = t+

(πt

1+ πt

)m = t+

(βRm − 1

βRm

)f (Rm), (52)

Suppose the fiscal authority sets g , t, and b. Then (52) determinesthe nominal interest rate Rm .

Given the interest rate, P0 is given by equation (50) asP0 = M0/f (Rm), where M0 is the initial money stock.



In subsequent periods, the price level is equal to Pt = P0 (βRm)t

where βRm = (1+ πt ) is the gross inflation rate. The nominal stockof money in each future period is endogenously determined byMt = Pt f (Rm).

Nominal interest rate must be such as to generate enough seigniorageto satisfy the government’s budget constraint.

In this example, even though the monetary authority has set M0

exogenously, the initial price level is determined by the need for fiscalsolvency.


The fiscal theory of the price level

Recently, a number of researchers have examined models in whichfiscal factors replace the money supply as the key determinant of theprice level.

Two ways fiscal policy might matter for the price level"

1 If fiscal variables affect real money demand, the price level will alsodepend on fiscal factors (see previous example).

2 If there are multiple price levels consistent with a given nominalquantity of money, fiscal policy may determine which of these is theequilibrium price level.


The fiscal theory of the price levelMultiple equilibria

Assume a constant nominal money supply M0.

Assume real interest rate fixed (or exogenous to money and prices).

The equilibrium between the real money supply and real moneydemand is

M0

Pt= g

(Pt+1Pt

), g ′ < 0.

Rewrite this as

Pt+1 = Ptg−1(M0

Pt

)≡ φ (Pt ) (53)

One (unique, stationary) solution is Pt+i = P∗ for all i ≥ 0, whereP∗ = M0g (1).

In this equilibrium, the quantity theory holds, P is proportional to M.



This may not be the only equilibrium.I Any price path starting at P0 = P ′ > P∗ is consistent with (53) andinvolves a speculative hyperinflation.

I Equilibria originating to the left of P∗ eventually violate a transversalitycondition since M/P is exploding as P → 0.)

I By itself, equation (53) is not suffi cient to unique determine theequilibrium value of P0, even though the nominal quantity of money isfixed.



P(t)

P(t+1)

P* P'

Figure: Equilibrium with a fixed nominal money supplyCarl E. Walsh (UC Santa Cruz) Gerzensee Study Center September 3-7, 2012 273 / 319

The fiscal theory of the price levelThe basic idea

If multiple equilibrium price levels are possible, does fiscal policy pindown a unique equilibrium?

Restrict analysis to perfect-foresight equilibria for simplicity.

Under standard assumptions, the household intertemporal budgetconstraint takes the form

dt +∞

∑i=0

λt ,t+i (yt+i − τt+i ) =∞

∑i=0

λt ,t+i

[ct+i +

(it+i

1+ it+i

)mdt+i

].

(54)

where λt ,t+i =i

∏j=1

(1

1+rt+j

)with λt ,t = 1.



The budget constraint for the government sector, in real terms,

gt + dt = τt +

(it

1+ it

)mt +

(1

1+ rt

)dt+1.

Recursively substituting for future values of dt+i , this budgetconstraint implies

dt +∞

∑i=0

λt ,t+i [gt+i − τt+i − st+i ] = limT→∞

λt ,t+T dT , (55)

where st = itmt/(1+ it ) is the government’s real seigniorage revenue.



DefinitionPolicy paths for (gt+i , τt+i , st+i , dt+i )i≥0 such that

dt +∞

∑i=0

λt ,t+i [gt+i − τt+i − st+i ] = limT→∞

λt ,t+T dT = 0

for all price paths pt+i , i ≥ 0 are called Ricardian policies.

DefinitionPolicy paths for (gt+i , τt+i , st+i , dt+i )i≥0 for which limT→∞ λt ,t+T dT maynot equal zero for all price paths are called non-Ricardian.



Now consider a perfect-foresight equilibrium; yt = ct + gt andmdt = mt . Substituting in (54) and rearranging yields

dt +∞

∑i=0

λt ,t+i

[gt+i − τt+i −

(it+i

1+ it+i

)mt+i

]= 0. (56)

Thus, an implication of the representative household’s optimizationproblem is that equation (56) must hold in equilibrium.


The fiscal theory of the price levelNon-Ricardian policies

Under a non-Ricardian policy, budget balance imposes an additionalcondition that must be satisfied in equilibrium.

This requirement can be written as

DtPt=

∞

∑i=0

λt ,t+i [τt+i + st+i − gt+i ] . (57)

At time t, the government’s outstanding nominal liabilities Dt arepredetermined by past policies.

Given the present discounted value of the government’s futuresurpluses, the only endogenous variable is the current price level Pt .The price level must adjust to ensure equation (57) is satisfied.



Suppose the real demand for money is given by

Mt

Pt= f (1+ it ). (58)

Equations (57) and (58) must both be satisfied in equilibrium.

Which two variables are determined jointed by these two equationsdepends on the assumptions that are made about fiscal and monetarypolicy.



Suppose the fiscal authority determines gt+i and τt+i for all i ≥ 0,and the monetary authority pegs the nominal rate of interest it+i = ıfor all i ≥ 0.Seigniorage is equal to ıf (1+ ı)/(1+ ı) and so is fixed by monetarypolicy. With this specification of monetary and fiscal policy, the rightside of

DtPt=

∞

∑i=0

λt ,t+i [τt+i + st+i − gt+i ]

is given.



Since Dt is predetermined at date t, this equation can be solved forthe equilibrium price level P∗t given by

P∗t =Dt

∑∞i=0 λt ,t+i [τt+i + st+i − gt+i ]

. (59)

The current nominal money supply is then determined by

Mt = P∗t f (1+ ı).

One property of this equilibrium is that changes in fiscal policy (g orτ) directly alter the equilibrium price level, even though seigniorage isunaffected.



In standard infinite horizon, representative agent models, a tax cut(current and future government expenditures unchanged) has noeffect on equilibrium (i.e., Ricardian equivalence holds) — thegovernment cannot engineer a permanent tax cut unless governmentexpenditures are also cut (in present value terms).

If budget balance holds only when evaluated at the equilibrium pricelevel, the government can plan a permanent tax cut. If it does, theprice level must rise to ensure the new, lower value of discountedsurpluses is again equal to the real value of government debt.


The fiscal theory of the price levelEmpirical evidence

Under the fiscal theory of the price level, intertemporal budgetbalance holds at the equilibrium value of the price level.

Under traditional theories of the price level, it holds for all values ofthe price level.

If we only observe equilibrium outcomes, it will be impossibleempirically to distinguish between the two theories.

I As Sims (1994) puts it, “Determinacy of the price level under anypolicy depends on the public’s beliefs about what the policy authoritywould do under conditions that are never observed in equilibrium.”


The fiscal policy in a neoclassical model

Suppose government imposes lump-sum taxes to finance exogenousstream of expenditures.

A rise in government expenditures has a negative wealth effect on theprivate sector because taxes must be raised.

This increases labor supply (so output increases) and reduces privateconsumption.

With distorting taxes, steady-state equilibrium output is reduced bytaxes.


The fiscal policy in a neoclassical model: example

If Yt = AtHαt , Ht = f

−1(Yt )⇒ Ht =(A−1t Yt

) 1a .If U = C 1−σ

t /(1− σ)−H1+φt /(1+ φ) . labor supply condition is

MRS = MPL or(A−1t Yt

) φa C−σ

t = αA1αt Y

1− 1α

t ⇒ C σt = αA

1+φαt Y

1− 1+φα

t

Goods market clearing condition is

Yt = Ct + Gt .

These imply

(Yt − Gt )−σ = αA1+φ

αt Y

1− 1+φα

t ⇒ dYtdGt

=1

1+( CY

) ( 1+αφσα

) < 1.


Fiscal policy in a new Keynesian model

Basic building blocks of the new Keynesian model:I Euler condition for optimal consumption:

ct = Etct+1 −(1σ

)(it − Etπt+1)

I Inflation adjustment:

πt = βEtπt+1 + κmct

I Marginal costs:mct = σct + ηnt − zt .



With government spending, Yt = Ct + Gt , or

ct =(YC

)yt −

(GC

)gt .

So model becomes(YC

)yt −

(GC

)gt =

(YC

)Etyt+1−

(GC

)Etgt+1−

(1σ

)(it − Etπt+1)

or

xt ≡ yt − yflext = Et (yt+1 − yflext+1)−(1σ

)(CY

)(it − Etπt+1)

−(GY

)(Etgt+1 − gt ) + Etyflext+1 − yflext ,



We can write this as

xt = Etxt+1 −(1σ

)(it − Etπt+1 − rnt ) ,

where

rnt = −σ

(GC

)(Etgt+1 − gt ) + σ

(Etyflext+1 − yflext

).

So fiscal shocks affect the IS curve.



Fiscal shocks can also affect inflation via marginal cost.

If Yt = eztNt , marginal cost is mct = σct + ηnt − zt , or

mct = σ

[yt −

(GY

)gt

]+ η(yt − zt )− zt ,

or

mct = (σ+ η)yt − σ

(GY

)gt − (1+ η)zt = (σ+ η)

(yt − yflext

),

where

yflext =σ(GY

)gt + (1+ η)ztσ+ η

.



IS equations becomes

xt = Etxt+1 −(1σ

)(it − Etπt+1 − rnt ) ,

which is the same as before, but rn depends on Etgt+1 − gt .Inflation equation becomes

πt = βEtπt+1 + κxt ,

where κ = κ(σ+ η), which is also the same as before.



But what about the loss function?I If loss function is π2t + λx x2t , then g only becomes a source of real rateshocks.

F In this case, optimal monetary policy will neutralize the effects of g onthe output gap and inflation.

F Flexible-price output is still affected via neoclassical channels.

I If loss is π2t + λx (yt − y∗t )2 with y∗t 6= yflext , then things are different.

So key issue is how fiscal policy affects the effi cient level of output y ∗.


Fiscal policy and monetary policyOptimal policy when flex-price and effi cient output differ

Policy problem is to

minEt∞

∑i=0

βi

π2t+i + λx (xt+i − x∗t+i )2 + θt+i (πt − βtπt+1 − κxt ),

where x ≡ y − yflex and x∗ = y ∗ − yflex .First order conditions (timeless perspective):

πt + θt − θt−1 = 0;

λx (xt − x∗t )− κθt = 0.

Combining,

πt = −(

λxκ

)∆ (xt − x∗t ) .


Fiscal policy and monetary policyOptimal policy when flex-price and effi cient output differ

Suppose y ∗t = (1+ η)zt/(σ+ η).

Then

yflext − y ∗t =σ(GY

)gt + (1+ η)ztσ+ η

− (1+ η)ztσ+ η

=

(σ

σ+ η

)(GY

)gt = γgt .

So FOC becomes

πt = −(

λxκ

)∆ (xt + γgt ) .

For optimal discretion,

πt = −(

λxκ

)(xt + γgt ) .


Fiscal policy at the ZLB

Suppose monetary policy cannot neutralize fiscal policy, for examplebecause interest rates are at the ZLB.

Argument is that at ZLB, fiscal policy is more powerful.I Erceg and Linde (2009), Eggertsson (2010).

At ZLB, old fashion Keynesian multiplier returns.I No crowding out since real interest rate does not rise.I Output expansion raises marginal cost and inflation, lowering the realinterest rate, so a form of crowding in occurs.


Output at the ZLB

Assume i = 0 and with probability µ economy is still at ZLB infollowing period. If economy exits ZLB, output and inflation are zero.

Equilibrium given by solutions to

yZ = µyZ −(1σ

)(−µπZ − rn

)⇒ yZ =

(1

1− µ

)(1σ

)(µπZ + rn

)πZ = βπZ + κyZ

where µyZ and µπZ are expected future output and inflation.


Adding taxes

Eggertsson (2010) —adds wage and sales taxes (plus other taxes)

Marginal cost:mct = (wt − pt )−mplt

w aftertaxt − pcpit ≈ (wt − τwt )− (pt + τst )

= ηnt + σ

[(YC

)yt −

(GC

)gt

]so

mct = η(yt − zt ) + σ

[(YC

)yt −

(GC

)gt

]+ τst + τwt − zt

=

[η + σ

(YC

)]yt − σ

(GC

)gt + τst + τwt − (1+ η)zt


Adding taxes:

The NKPC becomes

πt = βEtπt+1 + κ

[η + σ

(YC

)]yt − σ

(GY

)gt

+τst + τwt − (1+ η)zt

I Note that a rise in the wage tax or sales tax increases marginal costs.I So wage tax cuts are deflationary


Adding taxes:

Aggregate demand:

yt = Etyt −(1σ

)(it − Etπ

cpit+1 − rnt

)−(GY

)(Etgt+1 − gt )

= Etyt −(1σ

)(it − Etπt+1 − rnt )−

(GY

)(Etgt+1 − gt )

+

(1σ

)(Etτst+1 − τst )


Adding taxes: fiscal policy at the ZLBNow consider the case of the ZLB. Equilibrium is given by

πZt = βEtπt+1 + κ

[η + σ

(YC

)]yZt − κσ

(GY

)gt

+κ (τs + τw )− κ(1+ η)zt

yZt = Etyt+1 +(1σ

)(µπZt+1 + r

nt

)−(GY

)(Etgt+1 − gt )

+

(1σ

)(Etτst+1 − τs )

Expansionary policies:I temporary sales tax cut (boosts demand and inflation)I temporary rise in government spending (boosts demand but lowersinflation) (possibly)

I Expansionary policies: increase in wage tax (raises inflation anddemand).


Optimal taxation with distortionary taxesTax smoothing

Review of Barro’s result on optimal intertemporal taxation

Equalize marginal distortionary costs per dollar of revenue acrossI tax instrumentsI time


Distortionary taxes

If there are two tax rates τ1 and τ2, and distortions per dollar raisesare D(τ1, τ2),

∂D(τ1t , τ2t )∂τ1t

=∂D(τ1t , τ2t )

∂τ2t

and∂D(τ1t , τ2t )

∂τ1t= Et

[∂D(τ1t+1, τ2t+1)

∂τ1t+1

].

If D is quadratic, this second condition implies

τjt = Etτjt+1 ⇒ τjt+1 = τjt + ξ jt+1


Distortionary taxesTax smoothing

Tax rates follow random walks.

Level is set to finance expected present discounted value ofgovernment expenditures.

This is like a permanent income model of taxesI Taxes are like consumption — smooth them and use debt to financetransitory fluctuations in spending.


Debt and distortionary taxes

Suppose flexible-price equilibrium output depends negatively on taxon final output:

yflext = −ψτt + εt .

Inflation adjustment equation is

πt = βEtπt+1 + κ(yt − yflext

).

Suppose y ∗t is the effi cient or welfare maximizing output. Lossfunction is

Et∞

∑i=0

βi

π2t+i + λx (yt+i − y ∗t+i )2.


Government budget constraint

Assume level of nominal debt (as fraction of GDP) constant at d .

Let b be public debt as fraction of GDP.

ThenDt−1Pt

+ gt = τt + zt +(

11+ Rt

)DtPt,

where z represents (potential) lump-sum transfers.

This can be approximated by

bt−1 + gt = τt + zt + d (πt − βEtπt+1) + βbt

since 1/Pt ≈ (1− πt )/Pt−1 and (1+ Rt )−1 ≈ β(1− Etπt+1) sinceβ is equal to the steady-state inverse of the gross real rate of interest.


Ramsey optimal policy problem

The joint fiscal and monetary policy becomes

minπ,y ,τ

Et∞

∑i=0

βi

π2t+i + λx (yt+i − y ∗t+i )2

subject toπt = βEtπt+1 + κ (yt + ψτt − εt ) .

andbt−1 + gt = τt + zt + d (πt − βEtπt+1) + βbt



The first order conditions with respect to yt , πt bt , and τt (timelessperspective) are

λx (yt − y ∗t )− κθt = 0;

πt + θt − θt−1 − d(

ϕt − ϕt−1)= 0;

ϕt − Etϕt+1 = 0;

−κψθt − ϕt = 0,

where θ and ϕ are the Lagrangians on the inflation adjustmentequation and the budget constraint.



These FOCs imply that θt = (λx/κ)(yt − y ∗t ) and ϕt = −κψθt .

Hence, we can write the FOCs as

yt − y ∗t =(

κ

λx

)θt = −

(1

ψλx

)ϕt ;

πt =

(1

ψλx

)(λxκ+ κψd

)∆ϕt ;

ϕt = Etϕt+1.

Output gap and inflation are functions of the shadow cost ofgovernment budget resources and this cost is a random walk.


ImplicationsUnrestricted lump-sum taxes

Suppose z is unrestricted

First best — if there are no restrictions on z , optimal choice impliesϕt = 0, so πt = 0 and yt = yflext = y ∗t .

With y = y ∗,

yt = y ∗t = −ψτt + εt ⇒ τt = −(1ψ

)(y ∗t − εt ) ≡ τ∗t ,

where τ∗t is the tax (or subsidy) needed to ensure yflex = y ∗.

Fiscal policy used to stabilize output at y ∗t ; inflation kept equal tozero.


ImplicationsRestricted lump-sum taxes

Suppose lump-sum transfers are not available: zt ≡ 0.From FOC’s, ϕt = Etϕt+1, Et∆ϕt+1 = 0, so from

πt =

(1

ψλx

)(λxκ+ κψd

)∆ϕt

if follows that Etπt+1 = 0.Hence, the government’s budget constraint becomes

bt−1 + gt = τt + dπt + βbt

Solving forward,

(1− β)bt−1 = (1− β)dπt − ft + (1− β)∞

∑i=0

βiEt (τt+i − τ∗t+i ) ,

where

ft = (1− β)∞

∑i=0

βiEt (gt+i − τ∗t+i ).



The variable ft is the annuity value of government spending plus thesubsidy necessary to implement the first best.

Implications: restricted z

FOC for yt implies

yt − y ∗t = −ψτt + εt − y ∗t = −ψ(τt − τ∗t ) = −(1

ψλx

)ϕt .

This implies

Et (τt+i − τ∗t+i ) = (1/ψ2λx )Etϕt+i = (1/ψ2λx )ϕt = τt − τ∗t .



Hence,

(1− β)∞

∑i=0

βiEt (τt+i − τ∗t+i ) = (1− β)∞

∑i=0

βi (τt − τ∗t ) = τt − τ∗t .

Therefore,

ϕt = ψ2λx (τt − τ∗t ) =(ψ2λx

)[(1− β)bt−1 + ft − (1− β)dπt ]



Taking expectations as of t − 1:

ϕt−1 = Et−1ϕt =(ψ2λx

)[(1− β)bt−1 + Et−1ft ]

So

ϕt − ϕt−1 = ∆ϕt =(ψ2λx

)[(ft − Et−1ft )− (1− β)dπt ]

From FOCs,

πt =

(1

ψλx

)(λxκ+ κψd

)∆ϕt = A∆ϕt ;



Hence,

∆ϕt =(ψ2λx

)[(ft − Et−1ft )− (1− β)dA∆ϕt ]

=

(ψ2λx

1+ ψ2λx (1− β)dA

)(ft − Et−1ft )

= η (ft − Et−1ft )

where

η =

(ψ2λx

1+ ψ2λx (1− β)dA

)



Inflation, output, and the tax rate satisfy

πt = Aη (ft − Et−1ft )

∆ (yt − y ∗t ) = −(1

ψλx

)∆ϕt = −

(1

ψλx

)η (ft − Et−1ft )

yt − yt−1 = y ∗t − y ∗t−1 −(1

ψλx


τt − τt−1 = τ∗t − τ∗t−1 +

(1

ψ2λx



ImplicationsSummary

Inflation is a white noise process, depending on unexpected shifts inthe fiscal variable f .

Expected future inflation always equal to zero.

The output gap and the tax rate follow martingales plus a componentrelated to fluctuations in the first best level of output.

Both monetary and fiscal policy need to be used, but shocks are notcompletely stabilized.


Conclusions on monetary and fiscal policy

Integration of monetary and fiscal policy in optimizing models withnominal rigidities is new area of work.

Intertemporal budget constraint links monetary and fiscal decisions.

Fiscal theory reminds us that monetary policy cannot ignore fiscalpolicy.

New Keynesian framework can be expanding to study jointdetermination of optimal policy.


Summing upEvolving views: then

policy as systematic

...[equilibrium methods] will focus attention on the need to think ofpolicy as the choice of stable rules of the game, well understood byeconomic agents. Only in such a setting will economic theory helppredict the actions agents will choose to take. (Lucas and Sargent1978)

but unpredictable

.. the government countercyclical policy must itself beunforeseeable by private agents...while at the same time besystematically related to the state of the economy. Effectiveness, then,rests on the inability of private agents to recognize systematic patternsin monetary and fiscal policy. (Lucas and Sargent 1978).


Summing upEvolving views: now

policy as systematic and predictable

...the central bank’s stabilization goals can be most effectivelyachieved only to the extent that the central bank not only actsappropriately, but is also understood by the private sector topredictably act in a certain way. The ability to successfully steerprivate-sector expectations is favored by a decision procedure that isbased on a rule, since in this case the systematic character of thecentral bank’s actions can be most easily made apparent to the public.(Woodford 2003, p. 465)


Advanced Topics in Monetary Economics II1 - Study Center

Documents

Transcript of Advanced Topics in Monetary Economics II1 - Study Center